Binary relationFrom Wikipedia, the free encyclopedia

Chapter 1

Binary relation

“Relation (mathematics)" redirects here. For a more general notion of relation, see finitary relation. For a morecombinatorial viewpoint, see theory of relations. For other uses, see Relation § Mathematics.

In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is asubset of the Cartesian product A2 = A × A. More generally, a binary relation between two sets A and B is a subsetof A × B. The terms correspondence, dyadic relation and 2-place relation are synonyms for binary relation.An example is the "divides" relation between the set of prime numbers P and the set of integers Z, in which everyprime p is associated with every integer z that is a multiple of p (but with no integer that is not a multiple of p). Inthis relation, for instance, the prime 2 is associated with numbers that include −4, 0, 6, 10, but not 1 or 9; and theprime 3 is associated with numbers that include 0, 6, and 9, but not 4 or 13.Binary relations are used in many branches of mathematics to model concepts like "is greater than", "is equal to", and“divides” in arithmetic, "is congruent to" in geometry, “is adjacent to” in graph theory, “is orthogonal to” in linearalgebra and many more. The concept of function is defined as a special kind of binary relation. Binary relations arealso heavily used in computer science.A binary relation is the special case n = 2 of an n-ary relation R ⊆ A1 × … × An, that is, a set of n-tuples where thejth component of each n-tuple is taken from the jth domain Aj of the relation. An example for a ternary relation onZ×Z×Z is “lies between ... and ...”, containing e.g. the triples (5,2,8), (5,8,2), and (−4,9,−7).In some systems of axiomatic set theory, relations are extended to classes, which are generalizations of sets. Thisextension is needed for, among other things, modeling the concepts of “is an element of” or “is a subset of” in settheory, without running into logical inconsistencies such as Russell’s paradox.

1.1 Formal definition

A binary relation R is usually defined as an ordered triple (X, Y, G) where X and Y are arbitrary sets (or classes), andG is a subset of the Cartesian product X × Y. The sets X and Y are called the domain (or the set of departure) andcodomain (or the set of destination), respectively, of the relation, and G is called its graph.The statement (x,y) ∈ G is read "x is R-related to y", and is denoted by xRy or R(x,y). The latter notation correspondsto viewing R as the characteristic function on X × Y for the set of pairs of G.The order of the elements in each pair ofG is important: if a ≠ b, then aRb and bRa can be true or false, independentlyof each other. Resuming the above example, the prime 3 divides the integer 9, but 9 doesn't divide 3.A relation as defined by the triple (X, Y, G) is sometimes referred to as a correspondence instead.[1] In this case therelation from X to Y is the subset G of X × Y, and “from X to Y" must always be either specified or implied by thecontext when referring to the relation. In practice correspondence and relation tend to be used interchangeably.

2

1.2. SPECIAL TYPES OF BINARY RELATIONS 3

1.1.1 Is a relation more than its graph?

According to the definition above, two relations with identical graphs but different domains or different codomainsare considered different. For example, if G = {(1, 2), (1, 3), (2, 7)} , then (Z,Z, G) , (R,N, G) , and (N,R, G) arethree distinct relations, where Z is the set of integers and R is the set of real numbers.Especially in set theory, binary relations are often defined as sets of ordered pairs, identifying binary relations withtheir graphs. The domain of a binary relation R is then defined as the set of all x such that there exists at least oney such that (x, y) ∈ R , the range of R is defined as the set of all y such that there exists at least one x such that(x, y) ∈ R , and the field of R is the union of its domain and its range.[2][3][4]

A special case of this difference in points of view applies to the notion of function. Many authors insist on distin-guishing between a function’s codomain and its range. Thus, a single “rule,” like mapping every real number x tox2, can lead to distinct functions f : R → R and f : R → R+ , depending on whether the images under thatrule are understood to be reals or, more restrictively, non-negative reals. But others view functions as simply sets ofordered pairs with unique first components. This difference in perspectives does raise some nontrivial issues. As anexample, the former camp considers surjectivity—or being onto—as a property of functions, while the latter sees itas a relationship that functions may bear to sets.Either approach is adequate for most uses, provided that one attends to the necessary changes in language, notation,and the definitions of concepts like restrictions, composition, inverse relation, and so on. The choice between the twodefinitions usually matters only in very formal contexts, like category theory.

1.1.2 Example

Example: Suppose there are four objects {ball, car, doll, gun} and four persons {John, Mary, Ian, Venus}. Supposethat John owns the ball, Mary owns the doll, and Venus owns the car. Nobody owns the gun and Ian owns nothing.Then the binary relation “is owned by” is given as

R = ({ball, car, doll, gun}, {John, Mary, Ian, Venus}, {(ball, John), (doll, Mary), (car, Venus)}).

Thus the first element of R is the set of objects, the second is the set of persons, and the last element is a set of orderedpairs of the form (object, owner).The pair (ball, John), denoted by ₐ RJₒ means that the ball is owned by John.Two different relations could have the same graph. For example: the relation

({ball, car, doll, gun}, {John, Mary, Venus}, {(ball, John), (doll, Mary), (car, Venus)})

is different from the previous one as everyone is an owner. But the graphs of the two relations are the same.Nevertheless, R is usually identified or even defined as G(R) and “an ordered pair (x, y) ∈ G(R)" is usually denoted as"(x, y) ∈ R".

1.2 Special types of binary relations

Some important types of binary relations R between two sets X and Y are listed below. To emphasize that X and Ycan be different sets, some authors call such binary relations heterogeneous.[5][6]

Uniqueness properties:

• injective (also called left-unique[7]): for all x and z in X and y in Y it holds that if xRy and zRy then x = z. Forexample, the green relation in the diagram is injective, but the red relation is not, as it relates e.g. both x = −5and z = +5 to y = 25.

• functional (also called univalent[8] or right-unique[7] or right-definite[9]): for all x in X, and y and z in Yit holds that if xRy and xRz then y = z; such a binary relation is called a partial function. Both relations inthe picture are functional. An example for a non-functional relation can be obtained by rotating the red graphclockwise by 90 degrees, i.e. by considering the relation x=y2 which relates e.g. x=25 to both y=−5 and z=+5.

4 CHAPTER 1. BINARY RELATION

Example relations between real numbers. Red: y=x2. Green: y=2x+20.

• one-to-one (also written 1-to-1): injective and functional. The green relation is one-to-one, but the red is not.

Totality properties:

• left-total:[7] for all x in X there exists a y in Y such that xRy. For example R is left-total when it is a functionor a multivalued function. Note that this property, although sometimes also referred to as total, is differentfrom the definition of total in the next section. Both relations in the picture are left-total. The relation x=y2,obtained from the above rotation, is not left-total, as it doesn't relate, e.g., x = −14 to any real number y.

• surjective (also called right-total[7] or onto): for all y in Y there exists an x in X such that xRy. The greenrelation is surjective, but the red relation is not, as it doesn't relate any real number x to e.g. y = −14.

Uniqueness and totality properties:

1.3. RELATIONS OVER A SET 5

• A function: a relation that is functional and left-total. Both the green and the red relation are functions.

• An injective function: a relation that is injective, functional, and left-total.

• A surjective function or surjection: a relation that is functional, left-total, and right-total.

• A bijection: a surjective one-to-one or surjective injective function is said to be bijective, also known asone-to-one correspondence.[10] The green relation is bijective, but the red is not.

1.2.1 Difunctional

Less commonly encountered is the notion of difunctional (or regular) relation, defined as a relation R such thatR=RR−1R.[11]

To understand this notion better, it helps to consider a relation as mapping every element x∈X to a set xR = { y∈Y| xRy }.[11] This set is sometimes called the successor neighborhood of x in R; one can define the predecessorneighborhood analogously.[12] Synonymous terms for these notions are afterset and respectively foreset.[5]

A difunctional relation can then be equivalently characterized as a relation R such that wherever x1R and x2R have anon-empty intersection, then these two sets coincide; formally x1R ∩ x2R ≠ ∅ implies x1R = x2R.[11]

As examples, any function or any functional (right-unique) relation is difunctional; the converse doesn't hold. If oneconsiders a relation R from set to itself (X = Y), then if R is both transitive and symmetric (i.e. a partial equivalencerelation), then it is also difunctional.[13] The converse of this latter statement also doesn't hold.A characterization of difunctional relations, which also explains their name, is to consider two functions f: A → Cand g: B → C and then define the following set which generalizes the kernel of a single function as joint kernel: ker(f,g) = { (a, b) ∈ A × B | f(a) = g(b) }. Every difunctional relation R ⊆ A × B arises as the joint kernel of two functionsf: A → C and g: B → C for some set C.[14]

In automata theory, the term rectangular relation has also been used to denote a difunctional relation. This ter-minology is justified by the fact that when represented as a boolean matrix, the columns and rows of a difunctionalrelation can be arranged in such a way as to present rectangular blocks of true on the (asymmetric) main diagonal.[15]

Other authors however use the term “rectangular” to denote any heterogeneous relation whatsoever.[6]

1.3 Relations over a set

If X = Y then we simply say that the binary relation is over X, or that it is an endorelation over X.[16] In computerscience, such a relation is also called a homogeneous (binary) relation.[6][16][17] Some types of endorelations arewidely studied in graph theory, where they are known as simple directed graphs permitting loops.The set of all binary relations Rel(X) on a set X is the set 2X × X which is a Boolean algebra augmented with theinvolution of mapping of a relation to its inverse relation. For the theoretical explanation see Relation algebra.Some important properties of a binary relation R over a set X are:

• reflexive: for all x in X it holds that xRx. For example, “greater than or equal to” (≥) is a reflexive relation but“greater than” (>) is not.

• irreflexive (or strict): for all x in X it holds that not xRx. For example, > is an irreflexive relation, but ≥ is not.

• coreflexive: for all x and y in X it holds that if xRy then x = y. An example of a coreflexive relation is therelation on integers in which each odd number is related to itself and there are no other relations. The equalityrelation is the only example of a both reflexive and coreflexive relation.

The previous 3 alternatives are far from being exhaustive; e.g. the red relation y=x2 from theabove picture is neither irreflexive, nor coreflexive, nor reflexive, since it contains the pair(0,0), and (2,4), but not (2,2), respectively.

• symmetric: for all x and y in X it holds that if xRy then yRx. “Is a blood relative of” is a symmetric relation,because x is a blood relative of y if and only if y is a blood relative of x.

6 CHAPTER 1. BINARY RELATION

• antisymmetric: for all x and y in X, if xRy and yRx then x = y. For example, ≥ is anti-symmetric (so is >, butonly because the condition in the definition is always false).[18]

• asymmetric: for all x and y in X, if xRy then not yRx. A relation is asymmetric if and only if it is bothanti-symmetric and irreflexive.[19] For example, > is asymmetric, but ≥ is not.

• transitive: for all x, y and z inX it holds that if xRy and yRz then xRz. For example, “is ancestor of” is transitive,while “is parent of” is not. A transitive relation is irreflexive if and only if it is asymmetric.[20]

• total: for all x and y in X it holds that xRy or yRx (or both). This definition for total is different from left totalin the previous section. For example, ≥ is a total relation.

• trichotomous: for all x and y in X exactly one of xRy, yRx or x = y holds. For example, > is a trichotomousrelation, while the relation “divides” on natural numbers is not.[21]

• Right Euclidean: for all x, y and z in X it holds that if xRy and xRz, then yRz.

• Left Euclidean: for all x, y and z in X it holds that if yRx and zRx, then yRz.

• Euclidean: An Euclidean relation is both left and right Euclidean. Equality is a Euclidean relation because ifx=y and x=z, then y=z.

• serial: for all x in X, there exists y in X such that xRy. "Is greater than" is a serial relation on the integers. Butit is not a serial relation on the positive integers, because there is no y in the positive integers (i.e. the naturalnumbers) such that 1>y.[22] However, "is less than" is a serial relation on the positive integers, the rationalnumbers and the real numbers. Every reflexive relation is serial: for a given x, choose y=x. A serial relation canbe equivalently characterized as every element having a non-empty successor neighborhood (see the previoussection for the definition of this notion). Similarly an inverse serial relation is a relation in which every elementhas non-empty predecessor neighborhood.[12]

• set-like (or local): for every x in X, the class of all y such that yRx is a set. (This makes sense only if relationson proper classes are allowed.) The usual ordering < on the class of ordinal numbers is set-like, while its inverse> is not.

A relation that is reflexive, symmetric, and transitive is called an equivalence relation. A relation that is symmetric,transitive, and serial is also reflexive. A relation that is only symmetric and transitive (without necessarily beingreflexive) is called a partial equivalence relation.A relation that is reflexive, antisymmetric, and transitive is called a partial order. A partial order that is total is calleda total order, simple order, linear order, or a chain.[23] A linear order where every nonempty subset has a least elementis called a well-order.

1.4 Operations on binary relations

If R, S are binary relations over X and Y, then each of the following is a binary relation over X and Y :

• Union: R ∪ S ⊆ X × Y, defined as R ∪ S = { (x, y) | (x, y) ∈ R or (x, y) ∈ S }. For example, ≥ is the union of >and =.

• Intersection: R ∩ S ⊆ X × Y, defined as R ∩ S = { (x, y) | (x, y) ∈ R and (x, y) ∈ S }.

If R is a binary relation over X and Y, and S is a binary relation over Y and Z, then the following is a binary relationover X and Z: (see main article composition of relations)

• Composition: S ∘ R, also denoted R ; S (or more ambiguously R ∘ S), defined as S ∘ R = { (x, z) | there existsy ∈ Y, such that (x, y) ∈ R and (y, z) ∈ S }. The order of R and S in the notation S ∘ R, used here agrees withthe standard notational order for composition of functions. For example, the composition “is mother of” ∘ “isparent of” yields “is maternal grandparent of”, while the composition “is parent of” ∘ “is mother of” yields “isgrandmother of”.

1.4. OPERATIONS ON BINARY RELATIONS 7

A relation R on sets X and Y is said to be contained in a relation S on X and Y if R is a subset of S, that is, if x R yalways implies x S y. In this case, if R and S disagree, R is also said to be smaller than S. For example, > is containedin ≥.If R is a binary relation over X and Y, then the following is a binary relation over Y and X:

• Inverse or converse: R −1, defined as R −1 = { (y, x) | (x, y) ∈ R }. A binary relation over a set is equal to itsinverse if and only if it is symmetric. See also duality (order theory). For example, “is less than” (<) is theinverse of “is greater than” (>).

If R is a binary relation over X, then each of the following is a binary relation over X:

• Reflexive closure: R =, defined as R = = { (x, x) | x ∈ X } ∪ R or the smallest reflexive relation over X containingR. This can be proven to be equal to the intersection of all reflexive relations containing R.

• Reflexive reduction: R ≠, defined as R ≠ = R \ { (x, x) | x ∈ X } or the largest irreflexive relation over Xcontained in R.

• Transitive closure: R +, defined as the smallest transitive relation over X containing R. This can be seen to beequal to the intersection of all transitive relations containing R.

• Transitive reduction: R −, defined as a minimal relation having the same transitive closure as R.

• Reflexive transitive closure: R *, defined as R * = (R +) =, the smallest preorder containing R.

• Reflexive transitive symmetric closure: R ≡, defined as the smallest equivalence relation over X containingR.

1.4.1 Complement

If R is a binary relation over X and Y, then the following too:

• The complement S is defined as x S y if not x R y. For example, on real numbers, ≤ is the complement of >.

The complement of the inverse is the inverse of the complement.If X = Y, the complement has the following properties:

• If a relation is symmetric, the complement is too.

• The complement of a reflexive relation is irreflexive and vice versa.

• The complement of a strict weak order is a total preorder and vice versa.

The complement of the inverse has these same properties.

1.4.2 Restriction

The restriction of a binary relation on a set X to a subset S is the set of all pairs (x, y) in the relation for which x andy are in S.If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partialorder, total order, strict weak order, total preorder (weak order), or an equivalence relation, its restrictions are too.However, the transitive closure of a restriction is a subset of the restriction of the transitive closure, i.e., in generalnot equal. For example, restricting the relation "x is parent of y" to females yields the relation "x is mother ofthe woman y"; its transitive closure doesn't relate a woman with her paternal grandmother. On the other hand, thetransitive closure of “is parent of” is “is ancestor of"; its restriction to females does relate a woman with her paternalgrandmother.

8 CHAPTER 1. BINARY RELATION

Also, the various concepts of completeness (not to be confused with being “total”) do not carry over to restrictions.For example, on the set of real numbers a property of the relation "≤" is that every non-empty subset S of R with anupper bound in R has a least upper bound (also called supremum) in R. However, for a set of rational numbers thissupremum is not necessarily rational, so the same property does not hold on the restriction of the relation "≤" to theset of rational numbers.The left-restriction (right-restriction, respectively) of a binary relation between X and Y to a subset S of its domain(codomain) is the set of all pairs (x, y) in the relation for which x (y) is an element of S.

1.4.3 Algebras, categories, and rewriting systems

Various operations on binary endorelations can be treated as giving rise to an algebraic structure, known as relationalgebra. It should not be confused with relational algebra which deals in finitary relations (and in practice also finiteand many-sorted).For heterogenous binary relations, a category of relations arises.[6]

Despite their simplicity, binary relations are at the core of an abstract computation model known as an abstractrewriting system.

1.5 Sets versus classes

Certain mathematical “relations”, such as “equal to”, “member of”, and “subset of”, cannot be understood to be binaryrelations as defined above, because their domains and codomains cannot be taken to be sets in the usual systems ofaxiomatic set theory. For example, if we try to model the general concept of “equality” as a binary relation =, wemust take the domain and codomain to be the “class of all sets”, which is not a set in the usual set theory.In most mathematical contexts, references to the relations of equality, membership and subset are harmless becausethey can be understood implicitly to be restricted to some set in the context. The usual work-around to this problemis to select a “large enough” set A, that contains all the objects of interest, and work with the restriction =A instead of=. Similarly, the “subset of” relation ⊆ needs to be restricted to have domain and codomain P(A) (the power set ofa specific set A): the resulting set relation can be denoted ⊆A. Also, the “member of” relation needs to be restrictedto have domain A and codomain P(A) to obtain a binary relation ∈A that is a set. Bertrand Russell has shown thatassuming ∈ to be defined on all sets leads to a contradiction in naive set theory.Another solution to this problem is to use a set theory with proper classes, such as NBG or Morse–Kelley set theory,and allow the domain and codomain (and so the graph) to be proper classes: in such a theory, equality, membership,and subset are binary relations without special comment. (A minor modification needs to be made to the concept ofthe ordered triple (X, Y, G), as normally a proper class cannot be a member of an ordered tuple; or of course onecan identify the function with its graph in this context.)[24] With this definition one can for instance define a functionrelation between every set and its power set.

1.6 The number of binary relations

The number of distinct binary relations on an n-element set is 2n2 (sequence A002416 in OEIS):Notes:

• The number of irreflexive relations is the same as that of reflexive relations.

• The number of strict partial orders (irreflexive transitive relations) is the same as that of partial orders.

• The number of strict weak orders is the same as that of total preorders.

• The total orders are the partial orders that are also total preorders. The number of preorders that are neithera partial order nor a total preorder is, therefore, the number of preorders, minus the number of partial orders,minus the number of total preorders, plus the number of total orders: 0, 0, 0, 3, and 85, respectively.

• the number of equivalence relations is the number of partitions, which is the Bell number.

1.7. EXAMPLES OF COMMON BINARY RELATIONS 9

The binary relations can be grouped into pairs (relation, complement), except that for n = 0 the relation is its owncomplement. The non-symmetric ones can be grouped into quadruples (relation, complement, inverse, inverse com-plement).

1.7 Examples of common binary relations

• order relations, including strict orders:

• greater than• greater than or equal to• less than• less than or equal to• divides (evenly)• is a subset of

• equivalence relations:

• equality• is parallel to (for affine spaces)• is in bijection with• isomorphy

• dependency relation, a finite, symmetric, reflexive relation.

• independency relation, a symmetric, irreflexive relation which is the complement of some dependency relation.

1.8 See also

• Confluence (term rewriting)

• Hasse diagram

• Incidence structure

• Logic of relatives

• Order theory

• Triadic relation

1.9 Notes[1] Encyclopedic dictionary of Mathematics. MIT. 2000. pp. 1330–1331. ISBN 0-262-59020-4.

[2] Suppes, Patrick (1972) [originally published by D. van Nostrand Company in 1960]. Axiomatic Set Theory. Dover. ISBN0-486-61630-4.

[3] Smullyan, Raymond M.; Fitting, Melvin (2010) [revised and corrected republication of the work originally published in1996 by Oxford University Press, New York]. Set Theory and the Continuum Problem. Dover. ISBN 978-0-486-47484-7.

[4] Levy, Azriel (2002) [republication of the work published by Springer-Verlag, Berlin, Heidelberg and New York in 1979].Basic Set Theory. Dover. ISBN 0-486-42079-5.

[5] Christodoulos A. Floudas; Panos M. Pardalos (2008). Encyclopedia of Optimization (2nd ed.). Springer Science & BusinessMedia. pp. 299–300. ISBN 978-0-387-74758-3.

10 CHAPTER 1. BINARY RELATION

[6] Michael Winter (2007). Goguen Categories: A Categorical Approach to L-fuzzy Relations. Springer. pp. x–xi. ISBN978-1-4020-6164-6.

[7] Kilp, Knauer and Mikhalev: p. 3. The same four definitions appear in the following:

• Peter J. Pahl; Rudolf Damrath (2001). Mathematical Foundations of Computational Engineering: A Handbook.Springer Science & Business Media. p. 506. ISBN 978-3-540-67995-0.

• Eike Best (1996). Semantics of Sequential and Parallel Programs. Prentice Hall. pp. 19–21. ISBN 978-0-13-460643-9.

• Robert-Christoph Riemann (1999). Modelling of Concurrent Systems: Structural and Semantical Methods in the HighLevel Petri Net Calculus. Herbert Utz Verlag. pp. 21–22. ISBN 978-3-89675-629-9.

[8] Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, ISBN 978-0-521-76268-7, Chapt. 5

[9] Mäs, Stephan (2007), “Reasoning on Spatial Semantic Integrity Constraints”, Spatial Information Theory: 8th InternationalConference, COSIT 2007, Melbourne, Australia, September 19–23, 2007, Proceedings, Lecture Notes in Computer Science4736, Springer, pp. 285–302, doi:10.1007/978-3-540-74788-8_18

[10] Note that the use of “correspondence” here is narrower than as general synonym for binary relation.

[11] Chris Brink; Wolfram Kahl; Gunther Schmidt (1997). Relational Methods in Computer Science. Springer Science &Business Media. p. 200. ISBN 978-3-211-82971-4.

[12] Yao, Y. (2004). “Semantics of Fuzzy Sets in Rough Set Theory”. Transactions on Rough Sets II. Lecture Notes in ComputerScience 3135. p. 297. doi:10.1007/978-3-540-27778-1_15. ISBN 978-3-540-23990-1.

[13] William Craig (2006). Semigroups Underlying First-order Logic. American Mathematical Soc. p. 72. ISBN 978-0-8218-6588-0.

[14] Gumm, H. P.; Zarrad, M. (2014). “Coalgebraic Simulations and Congruences”. Coalgebraic Methods in Computer Science.Lecture Notes in Computer Science 8446. p. 118. doi:10.1007/978-3-662-44124-4_7. ISBN 978-3-662-44123-7.

[15] Julius Richard Büchi (1989). Finite Automata, Their Algebras and Grammars: Towards a Theory of Formal Expressions.Springer Science & Business Media. pp. 35–37. ISBN 978-1-4613-8853-1.

[16] M. E. Müller (2012). Relational Knowledge Discovery. Cambridge University Press. p. 22. ISBN 978-0-521-19021-3.

[17] Peter J. Pahl; Rudolf Damrath (2001). Mathematical Foundations of Computational Engineering: A Handbook. SpringerScience & Business Media. p. 496. ISBN 978-3-540-67995-0.

[18] Smith, Douglas; Eggen, Maurice; St. Andre, Richard (2006),ATransition to AdvancedMathematics (6th ed.), Brooks/Cole,p. 160, ISBN 0-534-39900-2

[19] Nievergelt, Yves (2002), Foundations of Logic and Mathematics: Applications to Computer Science and Cryptography,Springer-Verlag, p. 158.

[20] Flaška, V.; Ježek, J.; Kepka, T.; Kortelainen, J. (2007). Transitive Closures of Binary Relations I (PDF). Prague: Schoolof Mathematics – Physics Charles University. p. 1. Lemma 1.1 (iv). This source refers to asymmetric relations as “strictlyantisymmetric”.

[21] Since neither 5 divides 3, nor 3 divides 5, nor 3=5.

[22] Yao, Y.Y.; Wong, S.K.M. (1995). “Generalization of rough sets using relationships between attribute values” (PDF).Proceedings of the 2nd Annual Joint Conference on Information Sciences: 30–33..

[23] Joseph G. Rosenstein, Linear orderings, Academic Press, 1982, ISBN 0-12-597680-1, p. 4

[24] Tarski, Alfred; Givant, Steven (1987). A formalization of set theory without variables. American Mathematical Society. p.3. ISBN 0-8218-1041-3.

1.10 References• M. Kilp, U. Knauer, A.V. Mikhalev, Monoids, Acts and Categories: with Applications to Wreath Products andGraphs, De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, ISBN 3-11-015248-7.

• Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, ISBN 978-0-521-76268-7.

1.11. EXTERNAL LINKS 11

1.11 External links• Hazewinkel, Michiel, ed. (2001), “Binary relation”, Encyclopedia of Mathematics, Springer, ISBN 978-1-

55608-010-4

Chapter 2

Intransitivity

This article is about intransitivity in mathematics. For the linguistics sense, see Intransitive verb.

In mathematics, intransitivity (sometimes called nontransitivity) is a property of binary relations that are nottransitive relations. This may include any relation that is not transitive, or the stronger property of antitransitiv-ity, which describes a relation that is never transitive.

2.1 Intransitivity

A relation is transitive if, whenever it relates some A to some B, and that B to some C, it also relates that A to that C.Some authors call a relation intransitive if it is not transitive, i.e. (if the relation in question is named R )

¬ (∀a, b, c : aRb ∧ bRc ⇒ aRc) .

This statement is equivalent to

∃a, b, c : aRb ∧ bRc ∧ ¬(aRc)

For instance, in the food chain, wolves feed on deer, and deer feed on grass, but wolves do not feed on grass.[1] Thus,the feed on relation among life forms is intransitive, in this sense.Another example that does not involve preference loops arises in freemasonry: it may be the case that lodge Arecognizes lodge B, and lodge B recognizes lodge C, but lodge A does not recognize lodge C. Thus the recognitionrelation among Masonic lodges is intransitive.

2.2 Antitransitivity

Often the term intransitive is used to refer to the stronger property of antitransitivity.We just saw that the feed on relation is not transitive, but it still contains some transitivity: for instance: humans feedon rabbits, rabbits feed on carrots, and humans also feed on carrots.A relation is antitransitive if this never occurs at all, i.e.,

∀a, b, c : aRb ∧ bRc ⇒ ¬aRc

Many authors use the term intransitivity to mean antitransitivity.[2][3]

An example of an antitransitive relation: the defeated relation in knockout tournaments. If player A defeated playerB and player B defeated player C, A can have never played C, and therefore, A has not defeated C.

12

2.3. CYCLES 13

2.3 Cycles

The term intransitivity is often used when speaking of scenarios in which a relation describes the relative preferencesbetween pairs of options, and weighing several options produces a “loop” of preference:

• A is preferred to B

• B is preferred to C

• C is preferred to A

Rock, paper, scissors; Nontransitive dice; and Penney’s game are examples. Real combative relations of competingspecies [4] and strategies of individual animals [5] can be cyclic as well.Assuming no option is preferred to itself i.e. the relation is irreflexive, a preference relation with a loop is not transitive.For if it is, each option in the loop is preferred to each option, including itself. This can be illustrated for this exampleof a loop among A, B, and C. Assume the relation is transitive. Then, since A is preferred to B and B is preferred toC, also A is preferred to C. But then, since C is preferred to A, also A is preferred to A.Therefore such a preference loop (or "cycle") is known as an intransitivity.Notice that a cycle is neither necessary nor sufficient for a binary relation to be not transitive. For example, anequivalence relation possesses cycles but is transitive. Now, consider the relation “is an enemy of” and suppose thatthe relation is symmetric and satisfies the condition that for any country, any enemy of an enemy of the country isnot itself an enemy of the country. This is an example of an antitransitive relation that does not have any cycles. Inparticular, by virtue of being antitransitive the relation is not transitive.Finally, let us work with the example of rock, paper, scissors, calling the three options A, B, and C. Now, the relationover A, B, and C is “defeats” and the standard rules of the game are such that A defeats B, B defeats C, and C defeatsA. Furthermore, it is also true that B does not defeat A, C does not defeat B, and A does not defeat C. Finally, it isalso true that no option defeats itself. This information can be depicted in a table:The first argument of the relation is a row and the second one is a column. Ones indicate the relation holds, zeroindicates that it does not hold. Now, notice that the following statement is true for any pair of elements x and y drawn(with replacement) from the set {A, B, C}: If x defeats y, and y defeats z, then x does not defeat z. Hence the relationis antitransitive.Thus, a cycle is neither necessary nor sufficient for a binary relation to be antitransitive.

2.4 Occurrences in preferences• Intransitivity can occur under majority rule, in probabilistic outcomes of game theory, and in the Condorcet

voting method in which ranking several candidates can produce a loop of preference when the weights arecompared (see voting paradox). Intransitive dice demonstrate that probabilities are not necessarily transitive.

• In psychology, intransitivity often occurs in a person’s system of values (or preferences, or tastes), potentiallyleading to unresolvable conflicts.

• Analogously, in economics intransitivity can occur in a consumer’s preferences. This may lead to consumerbehaviour that does not conform to perfect economic rationality. In recent years, economists and philosophershave questioned whether violations of transitivity must necessarily lead to 'irrational behaviour' (see Anand(1993)).

2.5 Likelihood

It has been suggested that Condorcet voting tends to eliminate “intransitive loops” when large numbers of votersparticipate because the overall assessment criteria for voters balances out. For instance, voters may prefer candidateson several different units of measure such as by order of social consciousness or by order of most fiscally conservative.In such cases intransitivity reduces to a broader equation of numbers of people and the weights of their units ofmeasure in assessing candidates.

14 CHAPTER 2. INTRANSITIVITY

Such as:

• 30% favor 60/40 weighting between social consciousness and fiscal conservatism

• 50% favor 50/50 weighting between social consciousness and fiscal conservatism

• 20% favor a 40/60 weighting between social consciousness and fiscal conservatism

While each voter may not assess the units of measure identically, the trend then becomes a single vector on whichthe consensus agrees is a preferred balance of candidate criteria.

2.6 References[1] Wolves do eat grass - see Engel, Cindy (2003). Wild Health: Lessons in Natural Wellness from the Animal Kingdom

(paperback ed.). Houghton Mifflin. p. 141. ISBN 0-618-34068-8..

[2] Guide to Logic, Relations II

[3] IntransitiveRelation

[4] Kerr B., Riley M.A., Feldman M.W., & Bohannan B.J.M. (2002). Local dispersal promotes biodiversity in a real-life gameof rock–paper–scissors. Nature. 418(171-174)

[5] Leutwyler, K. (2000). Mating Lizards Play a Game of Rock-Paper-Scissors. Scientific American.

2.7 Further reading• Anand, P (1993). Foundations of Rational Choice Under Risk. Oxford: Oxford University Press..

• Poddiakov, A., & Valsiner, J. (2013). Intransitivity cycles and their transformations: How dynamically adaptingsystems function. In: L. Rudolph (Ed.). Qualitative mathematics for the social sciences: Mathematical modelsfor research on cultural dynamics. Abingdon, NY: Routledge. Pp. 343-391.

Chapter 3

Near sets

Figure 1. Descriptively, very near sets

In mathematics, near sets are either spatially close or descriptively close. Spatially close sets have nonempty intersection.In other words, spatially close sets are not disjoint sets, since they always have at least one element in common. De-scriptively close sets contain elements that have matching descriptions. Such sets can be either disjoint or non-disjointsets. Spatially near sets are also descriptively near sets.The underlying assumption with descriptively close sets is that such sets contain elements that have location andmeasurable features such as colour and frequency of occurrence. The description of the element of a set is definedby a feature vector. Comparison of feature vectors provides a basis for measuring the closeness of descriptively near

15

16 CHAPTER 3. NEAR SETS

Figure 2. Descriptively, minimally near sets

sets. Near set theory provides a formal basis for the observation, comparison, and classification of elements in setsbased on their closeness, either spatially or descriptively. Near sets offer a framework for solving problems based onhuman perception that arise in areas such as image processing, computer vision as well as engineering and scienceproblems.Near sets have a variety of applications in areas such as topology[37], pattern detection and classification[50], abstract al-gebra[51], mathematics in computer science[38], and solving a variety of problems based on human perception[42][82][47][52][56]

that arise in areas such as image analysis[54][14][46][17][18], image processing[40], face recognition[13], ethology[64], aswell as engineering and science problems[55][64][42][19][17][18]. From the beginning, descriptively near sets have provedto be useful in applications of topology[37], and visual pattern recognition [50], spanning a broad spectrum of applica-tions that include camouflage detection, micropaleontology, handwriting forgery detection, biomedical image analy-sis, content-based image retrieval, population dynamics, quotient topology, textile design, visual merchandising, andtopological psychology.As an illustration of the degree of descriptive nearness between two sets, consider an example of the Henry colourmodel for varying degrees of nearness between sets of picture elements in pictures (see, e.g.,[17] §4.3). The two pairsof ovals in Fig. 1 and Fig. 2 contain coloured segments. Each segment in the figures corresponds to an equivalenceclass where all pixels in the class have similar descriptions, i.e., picture elements with similar colours. The ovals inFig.1 are closer to each other descriptively than the ovals in Fig. 2.

3.1. HISTORY 17

3.1 History

It has been observed that the simple concept of nearness unifies various concepts of topological structures[20] inas-much as the category Near of all nearness spaces and nearness preserving maps contains categories sTop (symmetrictopological spaces and continuous maps[3]),Prox (proximity spaces and δ -maps[8][67]),Unif (uniform spaces and uni-formly continuous maps[81][77]) andCont (contiguity spaces and contiguity maps[24]) as embedded full subcategories[20][59].The categories εANear and εAMer are shown to be full supercategories of various well-known categories, in-cluding the category sTop of symmetric topological spaces and continuous maps, and the category Met∞ ofextended metric spaces and nonexpansive maps. The notation A ↪→ B reads categoryA is embedded in categoryB. The categories εAMer and εANear are supercategories for a variety of familiar categories[76] shown in Fig. 3.Let εANear denote the category of all ε -approach nearness spaces and contractions, and let εAMer denote thecategory of all ε -approach merotopic spaces and contractions.

Figure 3. Supercats

Among these familiar categories is sTop , the symmetric form of Top (see category of topological spaces), the cat-egory with objects that are topological spaces and morphisms that are continuous maps between them[1][32]. Met∞

with objects that are extended metric spaces is a subcategory of εAP (having objects ε -approach spaces and con-tractions) (see also[57][75]). Let ρX , ρY be extended pseudometrics on nonempty sets X,Y , respectively. The mapf : (X, ρX) −→ (Y, ρY ) is a contraction if and only if f : (X, νDρX

) −→ (Y, νDρY) is a contraction. For

nonempty subsets A,B ∈ 2X , the distance function Dρ : 2X × 2X −→ [0,∞] is defined by

Dρ(A,B) =

{inf {ρ(a, b) : a ∈ A, b ∈ B}, ifA and Bempty not are ,

∞, ifA or Bempty is .

Thus ε AP is embedded as a full subcategory in εANear by the functor F : εAP −→ εANear defined byF ((X, ρ)) = (X, νDρ) and F (f) = f . Then f : (X, ρX) −→ (Y, ρY ) is a contraction if and only if f :

18 CHAPTER 3. NEAR SETS

(X, νDρX) −→ (Y, νDρY

) is a contraction. Thus εAP is embedded as a full subcategory in εANear by thefunctor F : εAP −→ εANear defined by F ((X, ρ)) = (X, νDρ) and F (f) = f. Since the category Met∞

of extended metric spaces and nonexpansive maps is a full subcategory of εAP , therefore, εANear is also a fullsupercategory of Met∞ . The category εANear is a topological construct[76].

Figure 4. Frigyes Riesz, 1880-1956

The notions of near and far[A] in mathematics can be traced back to works by Johann Benedict Listing and FelixHausdorff. The related notions of resemblance and similarity can be traced back to J.H. Poincaré, who introducedsets of similar sensations (nascent tolerance classes) to represent the results of G.T. Fechner’s sensation sensitivityexperiments[10] and a framework for the study of resemblance in representative spaces as models of what he termedphysical continua[63][60][61]. The elements of a physical continuum (pc) are sets of sensations. The notion of a pcand various representative spaces (tactile, visual, motor spaces) were introduced by Poincaré in an 1894 article onthe mathematical continuum[63], an 1895 article on space and geometry[60] and a compendious 1902 book on science

3.2. NEARNESS OF SETS 19

and hypothesis[61] followed by a number of elaborations, e.g.,[62]. The 1893 and 1895 articles on continua (Pt. 1,ch. II) as well as representative spaces and geometry (Pt. 2, ch IV) are included as chapters in[61]. Later, F. Rieszintroduced the concept of proximity or nearness of pairs of sets at the International Congress of Mathematicians(ICM) in 1908[65].During the 1960s, E.C. Zeeman introduced tolerance spaces in modelling visual perception[83]. A.B. Sossinskyobserved in 1986[71] that the main idea underlying tolerance space theory comes from Poincaré, especially[60]. In2002, Z. Pawlak and J. Peters[B] considered an informal approach to the perception of the nearness of physical objectssuch as snowflakes that was not limited to spatial nearness. In 2006, a formal approach to the descriptive nearness ofobjects was considered by J. Peters, A. Skowron and J. Stepaniuk[C] in the context of proximity spaces[39][33][35][21].In 2007, descriptively near sets were introduced by J. Peters[D][E] followed by the introduction of tolerance nearsets[41][45]. Recently, the study of descriptively near sets has led to algebraic[22][51], topological and proximity space[37]

foundations of such sets.

3.2 Nearness of sets

The adjective near in the context of near sets is used to denote the fact that observed feature value differences ofdistinct objects are small enough to be considered indistinguishable, i.e., within some tolerance.The exact idea of closeness or 'resemblance' or of 'being within tolerance' is universal enough to appear, quite naturally,in almost any mathematical setting (see, e.g.,[66]). It is especially natural in mathematical applications: practicalproblems, more often than not, deal with approximate input data and only require viable results with a tolerable levelof error[71].The words near and far are used in daily life and it was an incisive suggestion of F. Riesz[65] that these intuitiveconcepts be made rigorous. He introduced the concept of nearness of pairs of sets at the ICM in Rome in 1908. Thisconcept is useful in simplifying teaching calculus and advanced calculus. For example, the passage from an intuitivedefinition of continuity of a function at a point to its rigorous epsilon-delta definition is sometime difficult for teachersto explain and for students to understand. Intuitively, continuity can be explained using nearness language, i.e., afunction f : R → R is continuous at a point c , provided points {x} near c go into points {f(x)} near f(c) . UsingRiesz’s idea, this definition can be made more precise and its contrapositive is the familiar definition[4][36].

3.3 Generalization of set intersection

From a spatial point of view, nearness (aka proximity) is considered a generalization of set intersection. For disjointsets, a form of nearness set intersection is defined in terms of a set of objects (extracted from disjoint sets) that havesimilar features within some tolerance (see, e.g., §3 in[80]). For example, the ovals in Fig. 1 are considered near eachother, since these ovals contain pairs of classes that display similar (visually indistinguishable) colours.

3.4 Efremovič proximity space

Let X denote a metric topological space that is endowed with one or more proximity relations and let 2X denote thecollection of all subsets of X . The collection 2X is called the power set of X .There are many ways to define Efremovič proximities on topological spaces (discrete proximity, standard proximity,metric proximity, Čech proximity, Alexandroff proximity, and Freudenthal proximity), For details, see § 2, pp. 93–94 in[6]. The focus here is on standard proximity on a topological space. For A,B ⊂ X , A is near B (denoted byA δ B ), provided their closures share a common point.The closure of a subset A ∈ 2X (denoted by cl(A) ) is the usual Kuratowski closure of a set[F], introduced in § 4, p.20[27], is defined by

cl(A) = {x ∈ X : D(x,A) = 0} , whereD(x,A) = inf {d(x, a) : a ∈ A} .

i.e. cl(A) is the set of all points x in X that are close to A ( D(x,A) is the Hausdorff distance (see § 22, p. 128,

20 CHAPTER 3. NEAR SETS

in[15]) between x and the set A and d(x, a) = |x− a| (standard distance)). A standard proximity relation is definedby

δ ={(A,B) ∈ 2X × 2X : cl(A) ∩ cl(B) ̸= ∅

}.

Whenever sets A and B have no points in common, the sets are farfrom each other (denoted A δ B ).The following EF-proximity[G] space axioms are given by Jurij Michailov Smirnov[67] based on what Vadim Arsenye-vič Efremovič introduced during the first half of the 1930s[8]. Let A,B,E ∈ 2X .

EF.1 If the set A is close to B , then B is close to A .

EF.2 A ∪B is close to E , if and only if, at least one of the sets A or B is close to E .

EF.3 Two points are close, if and only if, they are the same point.

EF.4 All sets are far from the empty set ∅ .

EF.5 For any two sets A and B which are far from each other, there exists C,D ∈ 2X , C ∪D = X , such that Ais far from C and B is far from D (Efremovič-axiom).

The pair (X, δ) is called an EF-proximity space. In this context, a space is a set with some added structure. Witha proximity space X , the structure of X is induced by the EF-proximity relation δ . In a proximity space X , theclosure of A in X coincides with the intersection of all closed sets that contain A .

Theorem 1[67] The closure of any set A in the proximity space X is the set of points x ∈ X that are close to A .

3.5 Visualization of EF-axiom

Let the set X be represented by the points inside the rectangular region in Fig. 5. Also, let A,B be any two non-intersection subsets (i.e. subsets spatially far from each other) inX , as shown in Fig. 5. LetCc = X\C (complementof the set C ). Then from the EF-axiom, observe the following:

A δ B,

B ⊂ C,

D = Cc,

X = D ∪ C,

A ⊂ D, hence, we can writeA δ B ⇒ A δ C and B δ D, for some C,D in X so that C ∪D = X. ■

3.6 Descriptive proximity space

Descriptively near sets were introduced as a means of solving classification and pattern recognition problems arisingfrom disjoint sets that resemble each other[44][43]. Recently, the connections between near sets in EF-spaces and nearsets in descriptive EF-proximity spaces have been explored in[53][48].Again, let X be a metric topological space and let Φ = {ϕ1, . . . , ϕn} a set of probe functions that represent featuresof each x ∈ X . The assumption made here is X contains non-abstract points that have measurable features such asgradient orientation. A non-abstract point has a location and features that can be measured (see § 3 in [26]).A probe function ϕ : X → R represents a feature of a sample point in X . The mapping Φ : X −→ Rn is definedby Φ(x) = (ϕ1(x), . . . , ϕn(x)) , where Rn is an n-dimensional real Euclidean vector space. Φ(x) is a feature vectorfor x , which provides a description of x ∈ X . For example, this leads to a proximal view of sets of picture pointsin digital images[48].

3.6. DESCRIPTIVE PROXIMITY SPACE 21

X

AB

CC c

Figure 5. Example of a descriptive EF-proximity relation between sets A,B , and Cc

To obtain a descriptive proximity relation (denoted by δΦ ), one first chooses a set of probe functions. LetQ : 2X −→2R

n be a mapping on a subset of 2X into a subset of 2Rn . For example, let A,B ∈ 2X and Q(A),Q(B) denotesets of descriptions of points in A,B , respectively. That is,

Q(A) = {Φ(a) : a ∈ A} ,Q(B) = {Φ(b) : b ∈ B} .

The expression A δΦ B reads A is descriptively near B . Similarly, A δΦ B reads A is descriptively far from B . Thedescriptive proximity of A and B is defined by

A δΦ B ⇔ Q(cl(A)) δ Q(cl(B)) ̸= ∅.

The descriptive intersection ∩Φ of A and B is defined by

A ∩Φ B = {x ∈ A ∪B : Q(A) δ Q(B)} .

That is, x ∈ A ∪B is in A ∩Φ B , provided Φ(x) = Φ(a) = Φ(b) for some a ∈ A, b ∈ B . Observe that A and Bcan be disjoint and yet A ∩Φ B can be nonempty. The descriptive proximity relation δΦ is defined by

δΦ ={(A,B) ∈ 2X × 2X : cl(A) ∩Φ cl(B) ̸= ∅

}.

Whenever sets A and B have no points with matching descriptions, the sets are descriptively far from each other(denoted by A δΦ B ).The binary relation δΦ is a descriptive EF-proximity, provided the following axioms are satisfied for A,B,E ⊂ X .

22 CHAPTER 3. NEAR SETS

dEF.1 If the set A is descriptively close to B , then B is descriptively close to A .

dEF.2 A ∪B is descriptively close to E , if and only if, at least one of the sets A or B is descriptively close to E .

dEF.3 Two points x, y ∈ X are descriptively close, if and only if, the description of x matches the description ofy .

dEF.4 All nonempty sets are descriptively far from the empty set ∅ .

dEF.5 For any two sets A and B which are descriptively far from each other, there exists C,D ∈ 2X , C ∪D = X, such that A is descriptively far from C and B is descriptively far from D (Descriptive Efremovič axiom).

The pair (X, δΦ) is called a descriptive proximity space.

3.7 Proximal relator spaces

A relator is a nonvoid family of relations R on a nonempty set X [72]. The pair (X,R) (also denoted X(R) ) iscalled a relator space. Relator spaces are natural generalizations of ordered sets and uniform spaces[73][74]}. With theintroduction of a family of proximity relations Rδ on X , we obtain a proximal relator space (X,Rδ) . For simplicity,we consider only two proximity relations, namely, the Efremovič proximity δ [8] and the descriptive proximity δΦ indefining the descriptive relatorRδΦ

[53][48]. The pair (X,RδΦ) is called a proximal relator space [49]. In this work, Xdenotes a metric topological space that is endowed with the relations in a proximal relator. With the introduction of(X,RδΦ) , the traditional closure of a subset (e.g., [9][7]) can be compared with the more recent descriptive closureof a subset.In a proximal relator space X , the descriptive closure of a set A (denoted by clΦ(A) ) is defined by

clΦ(A) = {x ∈ X : Φ(x)δQ(cl(A))} .

That is, x ∈ X is in the descriptive closure of A , provided the closure of Φ(x) and the closure of Q(cl(A)) have atleast one element in common.

Theorem 2 [50] The descriptive closure of any set A in the descriptive EF-proximity space (X,RδΦ) is the set ofpoints x ∈ X that are descriptively close to A .

Theorem 3 [50] Kuratowski closure of a setA is a subset of the descriptive closure ofA in a descriptive EF-proximityspace.

Theorem 4 [49] Let (X,RδΦ) be a proximal relator space, A ⊂ X . Then cl(A) ⊆ clΦ(A) .

Proof Let Φ(x) ∈ Q(X \ cl(A)) such that Φ(x) = Φ(a) for some a ∈ clA . Consequently, Φ(x) ∈ Q(clΦ(A)) .Hence, cl(A) ⊆ clΦ(A)

In a proximal relator space, EF-proximity δ leads to the following results for descriptive proximity δΦ .

Theorem 5 [49] Let (X,RδΦ) be a proximal relator space, A,B,C ⊂ X . Then

1 ◦

A δ B implies A δΦ B .

2 ◦

(A ∪B) δ C implies (A ∪B) δΦ C .

3 ◦

clA δ clB implies clA δΦ clB .

Proof

3.8. DESCRIPTIVE δ -NEIGHBOURHOODS 23

1 ◦

A δ B ⇔ A ∩B ̸= ∅ . For x ∈ A ∩B,Φ(x) ∈ Q(A) and Φ(x) ∈ Q(B) . Consequently, A δΦ B .

1◦ ⇒ 2◦

3 ◦

clA δ clB implies that clA and clA have at least one point in common. Hence, 1 o ⇒ 3o .

■

3.8 Descriptive δ -neighbourhoods

X

BE 1

E 2

A

X \E 2

Figure 6. Example depicting δ -neighbourhoods

In a pseudometric proximal relator space X , the neighbourhood of a point x ∈ X (denoted by Nx,ε ), for ε > 0 , isdefined by

Nx,ε = {y ∈ X : d(x, y) < ε} .

The interior of a set A (denoted by int(A) ) and boundary of A (denoted by bdy(A) ) in a proximal relator space Xare defined by

int(A) = {x ∈ X : Nx,ε ⊆ A} .

bdy(A) = cl(A) \ int(A).

24 CHAPTER 3. NEAR SETS

A set A has a natural strong inclusion in a set B associated with δ [5][6]} (denoted by A ≪δ B ), provided A ⊂ intB, i.e., A δ X \ intB ( A is far from the complement of intB ). Correspondingly, a set A has a descriptive stronginclusion in a set B associated with δΦ (denoted by A ≪Φ B ), provided Q(A) ⊂ Q(intB) , i.e., A δΦ X \ intB (Q(A) is far from the complement of intB ).Let ≪Φ be a descriptive δ -neighbourhood relation defined by

≪Φ ={(A,B) ∈ 2X × 2X : Q(A) ⊂ Q(intB)

}.

That is, A≪Φ B , provided the description of each a ∈ A is contained in the set of descriptions of the points b ∈ intB. Now observe that any A,B in the proximal relator space X such that A δΦ B have disjoint δΦ -neighbourhoods,i.e.,

A δΦ B ⇔ A≪Φ E1, B ≪Φ E2, for some E1, E2 ⊂ X (See Fig. 6).

Theorem 6 [50] Any two sets descriptively far from each other belong to disjoint descriptive δΦ -neighbourhoods ina descriptive proximity space X .

A consideration of strong containment of a nonempty set in another set leads to the study of hit-and-miss topologiesand the Wijsman topology[2].

3.9 Tolerance near sets

Let ε be a real number greater than zero. In the study of sets that are proximally near within some tolerance, the setof proximity relations RδΦ is augmented with a pseudometric tolerance proximity relation (denoted by δΦ,ε ) definedby

DΦ(A,B) = inf {d(Φ(a),Φ(a)) : Φ(a) ∈ Q(A),Φ(a) ∈ Q(B)} ,

d(Φ(a),Φ(a)) =∑n

i=1|ϕi(a)− ϕi(b)|,

δΦ,ε ={(A,B) ∈ 2X × 2X : |D(cl(A), cl(B))| < ε

}.

Let RδΦ,ε = RδΦ ∪{δΦ,ε} . In other words, a nonempty set equipped with the proximal relator RδΦ,ε has underlyingstructure provided by the proximal relator RδΦ and provides a basis for the study of tolerance near sets in X that arenear within some tolerance. Sets A,B in a descriptive pseudometric proximal relator space (X,RδΦ,ε) are tolerancenear sets (i.e., A δΦ,ε B ), provided

DΦ(A,B) < ε.

3.10 Tolerance classes and preclasses

Relations with the same formal properties as similarity relations of sensations considered by Poincaré[62] are nowadays,after Zeeman[83], called tolerance relations. A tolerance τ on a set O is a relation τ ⊆ O × O that is reflexive andsymmetric. In algebra, the term tolerance relation is also used in a narrow sense to denote reflexive and symmetricrelations defined on universes of algebras that are also compatible with operations of a given algebra, i.e., they aregeneralizations of congruence relations (see e.g.,[12]). In referring to such relations, the term algebraic tolerance orthe term algebraic tolerance relation is used. Transitive tolerance relations are equivalence relations. A set O togetherwith a tolerance τ is called a tolerance space (denoted (O, τ) ). A set A ⊆ O is a τ -preclass (or briefly preclasswhen τ is understood) if and only if for any x, y ∈ A , (x, y) ∈ τ .The family of all preclasses of a tolerance space is naturally ordered by set inclusion and preclasses that are maximalwith respect to set inclusion are called τ -classes or just classes, when τ is understood. The family of all classes ofthe space (O, τ) is particularly interesting and is denoted by Hτ (O) . The family Hτ (O) is a covering of O [58].

3.10. TOLERANCE CLASSES AND PRECLASSES 25

The work on similarity by Poincaré and Zeeman presage the introduction of near sets[44][43] and research on similarityrelations, e.g.,[79]. In science and engineering, tolerance near sets are a practical application of the study of sets thatare near within some tolerance. A tolerance ε ∈ (0,∞] is directly related to the idea of closeness or resemblance(i.e., being within some tolerance) in comparing objects. By way of application of Poincaré's approach in definingvisual spaces and Zeeman’s approach to tolerance relations, the basic idea is to compare objects such as image patchesin the interior of digital images.

3.10.1 Examples

Simple ExampleThe following simple example demonstrates the construction of tolerance classes from real data. Consider the 20objects in the table below with |Φ| = 1 .

Let a tolerance relation be defined as

∼=ε= {(x, y) ∈ O ×O : ∥ Φ(x)− Φ(y) ∥2≤ ε}

Then, setting ε = 0.1 gives the following tolerance classes:

H∼=ε(O) ={{x1, x8, x10, x11}, {x1, x9, x10, x11, x14},{x2, x7, x18, x19},{x3, x12, x17},{x4, x13, x20}, {x4, x18},{x5, x6, x15, x16}, {x5, x6, x15, x20},{x6, x13, x20}}.

Observe that each object in a tolerance class satisfies the condition ∥ Φ(x)−Φ(y) ∥2≤ ε , and that almost all of theobjects appear in more than one class. Moreover, there would be twenty classes if the indiscernibility relation wasused since there are no two objects with matching descriptions.Image Processing Example

Figure 7. Example of images that are near each other. (a) and (b) Images from the freely available LeavesDataset (see, e.g.,www.vision.caltech.edu/archive.html).

The following example provides an example based on digital images. Let a subimage be defined as a small subset ofpixels belonging to a digital image such that the pixels contained in the subimage form a square. Then, let the sets Xand Y respectively represent the subimages obtained from two different images, and let O = {X ∪ Y } . Finally, letthe description of an object be given by the Green component in the RGB color model. The next step is to find all

26 CHAPTER 3. NEAR SETS

the tolerance classes using the tolerance relation defined in the previous example. Using this information, toleranceclasses can be formed containing objects that have similar (within some small ε ) values for the Green componentin the RGB colour model. Furthermore, images that are near (similar) to each other should have tolerance classesdivided among both images (instead of a tolerance classes contained solely in one of the images). For example, thefigure accompanying this example shows a subset of the tolerance classes obtained from two leaf images. In thisfigure, each tolerance class is assigned a separate colour. As can be seen, the two leaves share similar toleranceclasses. This example highlights a need to measure the degree of nearness of two sets.

3.11 Nearness measure

Let (U,RδΦ,ε) denote a particular descriptive pseudometric EF-proximal relator space equipped with the proximityrelation δΦ,ε and with nonempty subsets X,Y ∈ 2U and with the tolerance relation ∼=Φ,ε defined in terms of a set ofprobes Φ and with ε ∈ (0,∞] , where

Figure 8. Examples of degree of nearness between two sets: (a) High degree of nearness, and (b) Low degree of nearness.

≃Φ,ε= {(x, y) ∈ U × U | |Φ(x)− Φ(y)| ≤ ε}.

Further, assume Z = X ∪ Y and let HτΦ,ε(Z) denote the family of all classes in the space (Z,≃Φ,ε) .Let A ⊆ X,B ⊆ Y . The distance D

tNM: 2U × 2U :−→ [0,∞] is defined by

DtNM

(X,Y ) =

{1− tNM(A,B), if X and Y are not empty,∞, if X or Y is empty,

where

tNM(A,B) =

( ∑C∈HτΦ,ε

(Z)

|C|

)−1

·∑

C∈HτΦ,ε(Z)

|C|min(|C ∩A|, |[C ∩B|)max(|C ∩A|, |C ∩B|)

.

The details concerning tNM are given in[14][16][17]. The idea behind tNM is that sets that are similar should havea similar number of objects in each tolerance class. Thus, for each tolerance class obtained from the covering ofZ = X ∪Y , tNM counts the number of objects that belong to X and Y and takes the ratio (as a proper fraction) oftheir cardinalities. Furthermore, each ratio is weighted by the total size of the tolerance class (thus giving importanceto the larger classes) and the final result is normalized by dividing by the sum of all the cardinalities. The range oftNM is in the interval [0,1], where a value of 1 is obtained if the sets are equivalent (based on object descriptions)and a value of 0 is obtained if they have no descriptions in common.As an example of the degree of nearness between two sets, consider figure below in which each image consists of twosets of objects, X and Y . Each colour in the figures corresponds to a set where all the objects in the class share the

3.12. NEAR SET EVALUATION AND RECOGNITION (NEAR) SYSTEM 27

same description. The idea behind tNM is that the nearness of sets in a perceptual system is based on the cardinalityof tolerance classes that they share. Thus, the sets in left side of the figure are closer (more near) to each other interms of their descriptions than the sets in right side of the figure.

3.12 Near set evaluation and recognition (NEAR) system

Figure 9. NEAR system GUI.

The Near set Evaluation and Recognition (NEAR) system, is a system developed to demonstrate practical applicationsof near set theory to the problems of image segmentation evaluation and image correspondence. It was motivatedby a need for a freely available software tool that can provide results for research and to generate interest in nearset theory. The system implements a Multiple Document Interface (MDI) where each separate processing task isperformed in its own child frame. The objects (in the near set sense) in this system are subimages of the images beingprocessed and the probe functions (features) are image processing functions defined on the subimages. The systemwas written in C++ and was designed to facilitate the addition of new processing tasks and probe functions. Currently,the system performs six major tasks, namely, displaying equivalence and tolerance classes for an image, performingsegmentation evaluation, measuring the nearness of two images, performing Content Based Image Retrieval (CBIR),and displaying the output of processing an image using a specific probe function.

3.13 Proximity System

The Proximity System is an application developed to demonstrate descriptive-based topological approaches to near-ness and proximity within the context of digital image analysis. The Proximity System grew out of the work of S.Naimpally and J. Peters on Topological Spaces. The Proximity System was written in Java and is intended to run intwo different operating environments, namely on Android smartphones and tablets, as well as desktop platforms run-

28 CHAPTER 3. NEAR SETS

Figure 10. The Proximity System.

ning the Java Virtual Machine. With respect to the desktop environment, the Proximity System is a cross-platformJava application for Windows, OSX, and Linux systems, which has been tested on Windows 7 and Debian Linuxusing the Sun Java 6 Runtime. In terms of the implementation of the theoretical approaches, both the Android andthe desktop based applications use the same back-end libraries to perform the description-based calculations, wherethe only differences are the user interface and the Android version has less available features due to restrictions onsystem resources.

3.14 See also• Alternative set theory

• Category:Mathematical relations

• Category:Topology

• Feature vector

• Proximity space

• Rough set

• Topology

3.15 Notes1. ^ J.R. Isbell observed that the notions near and far are important in a uniform space. Sets A,B are far

(uniformaly distal), provided the {A,B} is a discrete collection. A nonempty set U is a uniform neighbour-hood of a set A , provided the complement of U is far from U . See, §33 in [23]

2. ^ The intuition that led to the discovery of descriptively near sets is given in Pawlak, Z.;Peters, J.F. (2002,2007) “Jak blisko (How Near)". Systemy Wspomagania Decyzji I 57 (109)

3. ^ Descriptively near sets are introduced in[48]. The connections between traditional EF-proximity and descrip-tive EF-proximity are explored in [37].

3.16. REFERENCES 29

4. ^ Reminiscent of M. Pavel’s approach, descriptions of members of sets objects are defined relative to vectorsof values obtained from real-valued functions called probes. See, Pavel, M. (1993). Fundamentals of patternrecognition. 2nd ed. New York: Marcel Dekker, for the introduction of probe functions considered in thecontext of image registration.

5. ^ A non-spatial view of near sets appears in, C.J. Mozzochi, M.S. Gagrat, and S.A. Naimpally, Symmetricgeneralized topological structures, Exposition Press, Hicksville, NY, 1976., and, more recently, nearness ofdisjoint sets X and Y based on resemblance between pairs of elements x ∈ X, y ∈ Y (i.e. x and y havesimilar feature vectors ϕ(x),ϕ(y) and the norm ∥ ϕ(x)− ϕ(y) ∥p< ε ) See, e.g.,[43][42][53].

6. ^ The basic facts about closure of a set were first pointed out by M. Fréchet in[11], and elaborated by B. Knasterand C. Kuratowski in[25].

7. ^Observe that up to the 1970s, proximitymeant EF-proximity, since this is the one that was studied intensively.The pre-1970 work on proximity spaces is exemplified by the series of papers by J. M. Smirnov during thefirst half of the 1950s[68][67][69][70], culminating in the compendious collection of results by S.A. Naimpally andB.D. Warrack[34]. But in view of later developments, there is a need to distinguish between various proximities.A basic proximity or Čech-proximity was introduced by E. Čech during the late 1930s (see §25 A.1, pp. 439-440 in [78]). The conditions for the non-symmetric case for a proximity were introduced by S. Leader[28] andfor the symmetric case by M.W. Lodato[29][30][31].

3.16 References1. ^ Adámek, J.; Herrlich, H.; Strecker, G. E. (1990). Abstract and concrete categories. London: Wiley-

Interscience. pp. ix+482.

2. ^ Beer, G. (1993), “Topologies on closed and closed convex sets”, London, UK: Kluwer Academic Pub., pp.xi + 340pp. Missing or empty |title= (help)

3. ^ Bentley, H. L.; Colebunders, E.; Vandermissen, E. (2009), “A convenient setting for completions and func-tion spaces”, in Mynard, F.; Pearl, E., Contemporary Mathematics, Providence, RI: American MathematicalSociety, pp. 37–88 Missing or empty |title= (help)

4. ^ Cameron, P.; Hockingand, J. G.; Naimpally, S. A. (1974). “Nearness–a better approach to continuity andlimits”. American Mathematical Monthly 81 (7): 739–745. doi:10.2307/2319561.

5. ^ Di Concilio, A. (2008), “Action, uniformity and proximity”, in Naimpally, S. A.; Di Maio, G., Theory andApplications of Proximity, Nearness and Uniformity, Seconda Università di Napoli, Napoli: Prentice-Hall, pp.71–88 Missing or empty |title= (help)

6. ^ a b Di Concilio, A. (2009). “Proximity: A powerful tool in extension theory, function spaces, hyperspaces,boolean algebras and point-free geometry”. ContemporaryMathematics 486: 89–114. doi:10.1090/conm/486/09508.

7. ^ Devi, R.; Selvakumar, A.; Vigneshwaran, M. (2010). " (I, γ) -generalized semi-closed sets in topologicalspaces”. FILOMAT 24 (1): 97–100. doi:10.2298/fil1001097d.

8. ^ a b c Efremovič, V. A. (1952). “The geometry of proximity I (in Russian)". Mat. Sb. (N.S.) 31 (73): 189–200.

9. ^ Peters, J. F. (2008). “A note on a-open sets and e ∗ -sets”. FILOMAT 22 (1): 89–96.

10. ^ Fechner, G. T. (1966). Elements of Psychophysics, vol. I. London, UK: Hold, Rinehart & Winston. pp. H.E. Adler’s trans. of Elemente der Psychophysik, 1860.

11. ^ Fréchet, M. (1906). “Sur quelques points du calcul fonctionnel”. Rend. Circ. Mat. Palermo 22: 1–74.doi:10.1007/bf03018603.

12. ^ Grätzer, G.; Wenzel, G. H. (1989). “Tolerances, covering systems, and the axiom of choice”. ArchivumMathematicum 25 (1-2): 27–34.

13. ^ Gupta, S.; Patnaik, K. (2008). “Enhancing performance of face recognition systems by using near set ap-proach for selecting facial features”. Journal of Theoretical and Applied Information Technology 4 (5): 433–441.

30 CHAPTER 3. NEAR SETS

14. ^ a b Hassanien, A. E.; Abraham, A.; Peters, J. F.; Schaefer, G.; Henry, C. (2009). “Rough sets and near sets inmedical imaging: A review, IEEE”. Transactions on Information Technology in Biomedicine 13 (6): 955–968.doi:10.1109/TITB.2009.2017017.

15. ^ Hausdorff, F. (1914). Grundz¨uge der mengenlehre. Leipzig: Veit and Company. pp. viii + 476.

16. ^ Henry, C.; Peters, J. F. (2010). “Perception-based image classification, International”. Journal of IntelligentComputing and Cybernetics 3 (3): 410–430. doi:10.1108/17563781011066701.

17. ^ a b c d Henry, C. J. (2010), “Near sets: Theory and applications”, Ph.D. thesis, Dept. Elec. Comp. Eng., Uni.of MB, supervisor: J.F. Peters

18. ^ a b Henry, C.; Peters, J. F. (2011). “Arthritic hand-finger movement similarity measurements: Tolerance nearset approach”. Computational and Mathematical Methods in Medicine 2011: 1–14. doi:10.1155/2011/569898.

19. ^ Henry, C. J.; Ramanna, S. (2011). “Parallel Computation in Finding Near Neighbourhoods”. Lecture Notesin Computer Science,: 523–532.

20. ^ a b Herrlich, H. (1974). “A concept of nearness”. General Topology and its Applications 4: 191–212.doi:10.1016/0016-660x(74)90021-x.

21. ^ Hocking, J. G.; Naimpally, S. A. (2009), “Nearness—a better approach to continuity and limits”, AllahabadMathematical Society Lecture Note Series 3, Allahabad: The Allahabad Mathematical Society, pp. iv+66,ISBN 978-81-908159-1-8 Missing or empty |title= (help)

22. ^ Ïnan, E.; Öztürk, M. A. (2012). “Near groups on nearness approximation spaces”. Hacettepe Journal ofMathematics and Statistics 41 (4): 545–558.

23. ^ Isbell, J. R. (1964). Uniform spaces. Providence, Rhode Island: American Mathematical Society. pp. xi +175.

24. ^ Ivanova, V. M.; Ivanov, A. A. (1959). “Contiguity spaces and bicompact extensions of topological spaces(russian)". Dokl. Akad. Nauk SSSR 127: 20–22.

25. ^ Knaster, B.; Kuratowski, C. (1921). “Sur les ensembles connexes”. Fundamenta Mathematicae 2: 206–255.

26. ^ Kovár, M. M. (2011). “A new causal topology and why the universe is co-compact”. arXiv:1112.0817[math-ph].

27. ^ Kuratowski, C. (1958), “Topologie i”, Warsaw: Panstwowe Wydawnictwo Naukowe, pp. XIII + 494pp.Missing or empty |title= (help)

28. ^ Leader, S. (1967). “Metrization of proximity spaces”. Proceedings of the American Mathematical Society18: 1084–1088. doi:10.2307/2035803.

29. ^ Lodato, M. W. (1962), “On topologically induced generalized proximity relations”, Ph.D. thesis, RutgersUniversity

30. ^ Lodato, M. W. (1964). “On topologically induced generalized proximity relations I”. Proceedings of theAmerican Mathematical Society 15: 417–422. doi:10.2307/2034517.

31. ^ Lodato, M. W. (1966). “On topologically induced generalized proximity relations II”. Pacific Journal ofMathematics 17: 131–135.

32. ^ MacLane, S. (1971). Categories for the working mathematician. Berlin: Springer. pp. v+262pp.

33. ^ Mozzochi, C. J.; Naimpally, S. A. (2009), “Uniformity and proximity”, Allahabad Mathematical SocietyLecture Note Series 2, Allahabad: The Allahabad Mathematical Society, pp. xii+153, ISBN 978-81-908159-1-8 Missing or empty |title= (help)

34. ^ Naimpally, S. A. (1970). Proximity spaces. Cambridge, UK: Cambridge University Press. pp. x+128. ISBN978-0-521-09183-1.

35. ^ Naimpally, S. A. (2009). Proximity approach to problems in topology and analysis. Munich, Germany:Oldenbourg Verlag. pp. ix + 204. ISBN 978-3-486-58917-7.

3.16. REFERENCES 31

36. ^ Naimpally, S. A.; Peters, J. F. (2013). “Preservation of continuity”. Scientiae Mathematicae Japonicae 76(2): 1–7.

37. ^ a b c d Naimpally, S. A.; Peters, J. F. (2013). Topology with Applications. Topological Spaces via Near andFar. Singapore: World Scientific.

38. ^ Naimpally, S. A.; Peters, J. F.; Wolski, M. (2013). Near set theory and applications. Special Issue inMathematics in Computer Science 7. Berlin: Springer. p. 136.

39. ^ Naimpally, S. A.; Warrack, B. D. (1970), “Proximity spaces”, Cambridge Tract in Mathematics 59, Cam-bridge, UK: Cambridge University Press, pp. x+128 Missing or empty |title= (help)

40. ^ Pal, S. K.; Peters, J. F. (2010). Rough fuzzy image analysis. Foundations and methodologies. London, UK,:CRC Press, Taylor & Francis Group. ISBN 9781439803295.

41. ^ Peters, J. F. (2009). “Tolerance near sets and image correspondence”. International Journal of Bio-InspiredComputation 1 (4): 239–245. doi:10.1504/ijbic.2009.024722.

42. ^ a b c Peters, J. F.; Wasilewski, P. (2009). “Foundations of near sets”. Information Sciences 179 (18): 3091–3109. doi:10.1016/j.ins.2009.04.018.

43. ^ a b c Peters, J. F. (2007). “Near sets. General theory about nearness of objects”. Applied MathematicalSciences 1 (53): 2609–2629.

44. ^ a b Peters, J. F. (2007). “Near sets. Special theory about nearness of objects”. Fundamenta Informaticae 75(1-4): 407–433.

45. ^ Peters, J. F. (2010). “Corrigenda and addenda: Tolerance near sets and image correspondence”. InternationalJournal of Bio-Inspired Computation 2 (5): 310–318. doi:10.1504/ijbic.2010.036157.

46. ^ Peters, J. F. (2011), “How near are Zdzisław Pawlak’s paintings? Merotopic distance between regions ofinterest”, in Skowron, A.; Suraj, S., Intelligent Systems Reference Library volume dedicated to Prof. ZdzisławPawlak, Berlin: Springer, pp. 1–19 Missing or empty |title= (help)

47. ^ Peters, J. F. (2011), “Sufficiently near sets of neighbourhoods”, in Yao, J. T.; Ramanna, S.; Wang, G. et al.,Lecture Notes in Artificial Intelligence 6954, Berlin: Springer, pp. 17–24 Missing or empty |title= (help);

48. ^ a b c d Peters, J. F. (2013). “Near sets: An introduction”. Mathematics in Computer Science 7 (1): 3–9.doi:10.1007/s11786-013-0149-6.

49. ^ a b c Peters, J. F. (2014). “Proximal relator spaces”. FILOMAT : 1–5 (in press).

50. ^ a b c d e Peters, J. F. (2014). Topology of Digital Images. Visual Pattern Discovery in Proximity Spaces 63.Springer. p. 342. ISBN 978-3-642-53844-5.

51. ^ a b Peters, J. F.; İnan, E.; Öztürk, M. A. (2014). “Spatial and descriptive isometries in proximity spaces”.General Mathematics Notes 21 (2): 125–134.

52. ^ Peters, J. F.; Naimpally, S. A. (2011). “Approach spaces for near families”. General Mathematics Notes 2(1): 159–164.

53. ^ a b c Peters, J. F.; Naimpally, S. A. (2011). General Mathematics Notes 2 (1): 159–164. Missing or empty|title= (help)

54. ^ Peters, J. F.; Puzio, L. (2009). “Image analysis with anisotropic wavelet-based nearness measures”. Interna-tional Journal of Computational Intelligence Systems 2 (3): 168–183. doi:10.1016/j.ins.2009.04.018.

55. ^ Peters, J. F.; Shahfar, S.; Ramanna, S.; Szturm, T. (2007), “Biologically-inspired adaptive learning: A nearset approach”, Frontiers in the Convergence of Bioscience and Information Technologies, Korea Missing orempty |title= (help)

56. ^ Peters, J. F.; Tiwari, S. (2011). “Approach merotopies and near filters. Theory and application”. GeneralMathematics Notes 3 (1): 32–45.

57. ^ Peters, J. F.; Tiwari, S. (2011). “Approach merotopies and near filters. Theory and application”. GeneralMathematics Notes 3 (1): 32–45.

32 CHAPTER 3. NEAR SETS

58. ^ Peters, J. F.; Wasilewski, P. (2012). “Tolerance spaces: Origins, theoretical aspects and applications”. In-formation Sciences 195: 211–225. doi:10.1016/j.ins.2012.01.023.

59. ^ Picado, J. “Weil nearness spaces”. Portugaliae Mathematica 55 (2): 233–254.

60. ^ a b c Poincaré, J. H. (1895). “L'espace et la géomètrie”. Revue de m'etaphysique et de morale 3: 631–646.

61. ^ a b c Poincaré, J. H. (1902). “Sur certaines surfaces algébriques; troisième complément 'a l'analysis situs”.Bulletin de la Société de France 30: 49–70.

62. ^ a b Poincaré, J. H. (1913 & 2009). Dernières pensées, trans. by J.W. Bolduc as Mathematics and science: Lastessays. Paris & NY: Flammarion & Kessinger. Check date values in: |date= (help)

63. ^ a b Poincaré, J. H. (1894). “Sur la nature du raisonnement mathématique”. Revue de méaphysique et demorale 2: 371–384.

64. ^ a b Ramanna, S.; Meghdadi, A. H. (2009). “Measuring resemblances between swarm behaviours: A percep-tual tolerance near set approach”. Fundamenta Informaticae 95 (4): 533–552. doi:10.3233/FI-2009-163.

65. ^ a b Riesz, F. (1908). “Stetigkeitsbegriff und abstrakte mengenlehre”. Atti del IV Congresso Internazionale deiMatematici II: 182–109.

66. ^ Shreider, J. A. (1975). Equality, resemblance, and order. Russia: Mir Publishers. p. 279.

67. ^ a b c d Smirnov, J. M. (1952). “On proximity spaces”. Mat. Sb. (N.S.) 31 (73): 543–574 (English translation:Amer. Math. Soc. Trans. Ser. 2, 38, 1964, 5–35).

68. ^ Smirnov, J. M. (1952). “On proximity spaces in the sense of V.A. Efremovič". Math. Sb. (N.S.) 84:895–898, English translation: Amer. Math. Soc. Trans. Ser. 2, 38, 1964, 1–4.

69. ^ Smirnov, J. M. (1954). “On the completeness of proximity spaces. I.”. Trudy Moskov. Mat. Obšč 3:271–306, English translation: Amer. Math. Soc. Trans. Ser. 2, 38, 1964, 37–74.

70. ^ Smirnov, J. M. (1955). “On the completeness of proximity spaces. II.”. Trudy Moskov. Mat. Obšč 4:421–438, English translation: Amer. Math. Soc. Trans. Ser. 2, 38, 1964, 75–94.

71. ^ a b Sossinsky, A. B. (1986). “Tolerance space theory and some applications”. Acta Applicandae Mathemati-cae: An International Survey Journal on Applying Mathematics and Mathematical Applications 5 (2): 137–167.doi:10.1007/bf00046585.

72. ^ Száz, Á. (1997). “Uniformly, proximally and topologically compact relators”. Mathematica Pannonica 8 (1):103–116.

73. ^ Száz, Á. (1987). “Basic tools and mild continuities in relator spaces”. Acta Mathematica Hungarica 50:177–201. doi:10.1007/bf01903935.

74. ^ Száz, Á (2000). “An extension of Kelley’s closed relation theorem to relator spaces”. FILOMAT 14: 49–71.

75. ^ Tiwari, S. (2010), “Some aspects of general topology and applications. Approach merotopic structures andapplications”, Ph.D. thesis, Dept. of Math., Allahabad (U.P.), India, supervisor: M. khare

76. ^ a b Tiwari, S.; Peters, J. F. (2013). “A new approach to the study of extended metric spaces”. MathematicaAeterna 3 (7): 565–577.

77. ^ Tukey, J. W. (1940), “Convergence and uniformity in topology”, Annals of Mathematics Studies AM–2,Princeton, NJ: Princeton Univ. Press, p. 90 Missing or empty |title= (help)

78. ^ Čech, E. (1966). Topological spaces, revised ed. by Z. Frolik and M. Katětov. London: John Wiley & Sons.p. 893.

79. ^ Wasilewski, P. (2004), “On selected similarity relations and their applications into cognitive science”, Ph.D.thesis, Dept. Logic

80. ^ Wasilewski, P.; Peters, J. F.; Ramanna, S. (2011). “Perceptual tolerance intersection”. Transactions onRough Sets XIII: 159–174.

3.17. FURTHER READING 33

81. ^ Weil, A. (1938), “Sur les espaces à structure uniforme et sur la topologie générale”, Actualités scientifiqueet industrielles, Paris: Harmann & cie Missing or empty |title= (help)

82. ^ Wolski, M. (2010). “Perception and classification. A note on near sets and rough sets”. Fundamenta Infor-maticae 101: 143–155.

83. ^ a b Zeeman, E. C. (1962), “The topology of the brain and visual perception”, in Fort, Jr., M. K., Topologyof 3-Manifolds and Related Topics, University of Georgia Institute Conference Proceedings (1962): Prentice-Hall, pp. 240–256 Missing or empty |title= (help)

3.17 Further reading• Naimpally, S. A.; Peters, J. F. (2013). Topology with Applications. Topological Spaces via Near and Far.

World Scientific Publishing . Co. Pte. Ltd. ISBN 978-981-4407-65-6.

• Naimpally, S. A.; Peters, J. F.; Wolski, M. (2013), "Near Set Theory and Applications", Mathematics inComputer Science 7 (1), Berlin: Springer Missing or empty |title= (help)

• Peters, J. F. (2014), "Topology of Digital Images. Visual Pattern Discovery in Proximity Spaces", IntelligentSystems Reference Library 63, Berlin: Springer Missing or empty |title= (help)

• Henry, C. J.; Peters, J. F. (2012), "Near set evaluation and recognition (NEAR) system V3.0", UM CI Labo-ratory Technical Report No. TR-2009-015, Computational Intelligence Laboratory, University of ManitobaMissing or empty |title= (help)

• Concilio, A. Di (2014). “Proximity: A powerful tool in extension theory, function spaces, hyperspaces, booleanalgebras and point-free geometry”. Computational Intelligence Laboratory, University of Manitoba. UM CILaboratory Technical Report No. TR-2009-021.

• Peters, J. F.; Naimpally, S. A. (2012). “Applications of near sets” (PDF).Notices of the AmericanMathematicalSociety 59 (4): 536–542. CiteSeerX: 10 .1 .1 .371 .7903.

Chapter 4

Preorder

Not to be confused with Pre-order.This article is about binary relations. For the graph vertex ordering, see Depth-first search. For other uses, seePreorder (disambiguation).“Quasiorder” redirects here. For irreflexive transitive relations, see strict order.

In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive.All equivalence relations and (non-strict) partial orders are preorders, but preorders are more general.The name 'preorder' comes from the idea that preorders (that are not partial orders) are 'almost' (partial) orders,but not quite; they're neither necessarily anti-symmetric nor symmetric. Because a preorder is a binary relation,the symbol ≤ can be used as the notational device for the relation. However, because they are not necessarily anti-symmetric, some of the ordinary intuition associated to the symbol ≤ may not apply. On the other hand, a preordercan be used, in a straightforward fashion, to define a partial order and an equivalence relation. Doing so, however, isnot always useful or worthwhile, depending on the problem domain being studied.In words, when a ≤ b, one may say that b covers a or that a precedes b, or that b reduces to a. Occasionally, thenotation ← or ≲ is used instead of ≤.To every preorder, there corresponds a directed graph, with elements of the set corresponding to vertices, and theorder relation between pairs of elements corresponding to the directed edges between vertices. The converse is nottrue: most directed graphs are neither reflexive nor transitive. In general, the corresponding graphs may containcycles. A preorder that is antisymmetric no longer has cycles; it is a partial order, and corresponds to a directedacyclic graph. A preorder that is symmetric is an equivalence relation; it can be thought of as having lost the directionmarkers on the edges of the graph. In general, a preorder may have many disconnected components.

4.1 Formal definition

Consider some set P and a binary relation ≤ on P. Then ≤ is a preorder, or quasiorder, if it is reflexive and transitive,i.e., for all a, b and c in P, we have that:

a ≤ a (reflexivity)if a ≤ b and b ≤ c then a ≤ c (transitivity)

A set that is equipped with a preorder is called a preordered set (or proset).[1]

If a preorder is also antisymmetric, that is, a ≤ b and b ≤ a implies a = b, then it is a partial order.On the other hand, if it is symmetric, that is, if a ≤ b implies b ≤ a, then it is an equivalence relation.Equivalently, the notion of a preordered set P can be formulated in a categorical framework as a thin category, i.e.as a category with at most one morphism from an object to another. Here the objects correspond to the elementsof P, and there is one morphism for objects which are related, zero otherwise. Alternately, a preordered set can beunderstood as an enriched category, enriched over the category 2 = (0→1).

34

4.2. EXAMPLES 35

A preordered class is a class equipped with a preorder. Every set is a class and so every preordered set is a preorderedclass.

4.2 Examples• The reachability relationship in any directed graph (possibly containing cycles) gives rise to a preorder, wherex ≤ y in the preorder if and only if there is a path from x to y in the directed graph. Conversely, every preorderis the reachability relationship of a directed graph (for instance, the graph that has an edge from x to y for everypair (x, y) with x ≤ y). However, many different graphs may have the same reachability preorder as each other.In the same way, reachability of directed acyclic graphs, directed graphs with no cycles, gives rise to partiallyordered sets (preorders satisfying an additional anti-symmetry property).

• Every finite topological space gives rise to a preorder on its points, in which x≤ y if and only if x belongs to everyneighborhood of y, and every finite preorder can be formed as the specialization preorder of a topological spacein this way. That is, there is a 1-to-1 correspondence between finite topologies and finite preorders. However,the relation between infinite topological spaces and their specialization preorders is not 1-to-1.

• A net is a directed preorder, that is, each pair of elements has an upper bound. The definition of convergencevia nets is important in topology, where preorders cannot be replaced by partially ordered sets without losingimportant features.

• The relation defined by x ≤ y if f(x) ≤ f(y) , where f is a function into some preorder.

• The relation defined by x ≤ y if there exists some injection from x to y. Injection may be replaced by surjection,or any type of structure-preserving function, such as ring homomorphism, or permutation.

• The embedding relation for countable total orderings.

• The graph-minor relation in graph theory.

• A category with at most one morphism from any object x to any other object y is a preorder. Such categoriesare called thin. In this sense, categories “generalize” preorders by allowing more than one relation betweenobjects: each morphism is a distinct (named) preorder relation.

In computer science, one can find examples of the following preorders.

• Many-one and Turing reductions are preorders on complexity classes.

• The subtyping relations are usually preorders.

• Simulation preorders are preorders (hence the name).

• Reduction relations in abstract rewriting systems.

• The encompassment preorder on the set of terms, defined by s≤t if a subterm of t is a substitution instance ofs.

Example of a total preorder:

• Preference, according to common models.

4.3 Uses

Preorders play a pivotal role in several situations:

• Every preorder can be given a topology, the Alexandrov topology; and indeed, every preorder on a set is inone-to-one correspondence with an Alexandrov topology on that set.

• Preorders may be used to define interior algebras.

• Preorders provide the Kripke semantics for certain types of modal logic.

36 CHAPTER 4. PREORDER

4.4 Constructions

Every binary relation R on a set S can be extended to a preorder on S by taking the transitive closure and reflexiveclosure, R+=. The transitive closure indicates path connection in R: x R+ y if and only if there is an R-path from x toy.Given a preorder ≲ on S one may define an equivalence relation ~ on S such that a ~ b if and only if a ≲ b and b ≲a. (The resulting relation is reflexive since a preorder is reflexive, transitive by applying transitivity of the preordertwice, and symmetric by definition.)Using this relation, it is possible to construct a partial order on the quotient set of the equivalence, S / ~, the set ofall equivalence classes of ~. Note that if the preorder is R+=, S / ~ is the set of R-cycle equivalence classes: x ∈ [y]if and only if x = y or x is in an R-cycle with y. In any case, on S / ~ we can define [x] ≤ [y] if and only if x ≲ y.By the construction of ~, this definition is independent of the chosen representatives and the corresponding relationis indeed well-defined. It is readily verified that this yields a partially ordered set.Conversely, from a partial order on a partition of a set S one can construct a preorder on S. There is a 1-to-1 corre-spondence between preorders and pairs (partition, partial order).For a preorder " ≲ ", a relation "<" can be defined as a < b if and only if (a≲ b and not b≲ a), or equivalently, usingthe equivalence relation introduced above, (a ≲ b and not a ~ b). It is a strict partial order; every strict partial ordercan be the result of such a construction. If the preorder is anti-symmetric, hence a partial order "≤", the equivalenceis equality, so the relation "<" can also be defined as a < b if and only if (a ≤ b and a ≠ b).(Alternatively, for a preorder " ≲ ", a relation "<" can be defined as a < b if and only if (a ≲ b and a ≠ b). The resultis the reflexive reduction of the preorder. However, if the preorder is not anti-symmetric the result is not transitive,and if it is, as we have seen, it is the same as before.)Conversely we have a ≲ b if and only if a < b or a ~ b. This is the reason for using the notation " ≲ "; "≤" can beconfusing for a preorder that is not anti-symmetric, it may suggest that a ≤ b implies that a < b or a = b.Note that with this construction multiple preorders " ≲ " can give the same relation "<", so without more information,such as the equivalence relation, " ≲ " cannot be reconstructed from "<". Possible preorders include the following:

• Define a ≤ b as a < b or a = b (i.e., take the reflexive closure of the relation). This gives the partial orderassociated with the strict partial order "<" through reflexive closure; in this case the equivalence is equality, sowe don't need the notations ≲ and ~.

• Define a≲ b as “not b < a" (i.e., take the inverse complement of the relation), which corresponds to defining a~ b as “neither a < b nor b < a"; these relations ≲ and ~ are in general not transitive; however, if they are, ~ isan equivalence; in that case "<" is a strict weak order. The resulting preorder is total, that is, a total preorder.

4.5 Number of preorders

As explained above, there is a 1-to-1 correspondence between preorders and pairs (partition, partial order). Thus thenumber of preorders is the sum of the number of partial orders on every partition. For example:

• for n=3:

• 1 partition of 3, giving 1 preorder• 3 partitions of 2+1, giving 3 × 3 = 9 preorders• 1 partition of 1+1+1, giving 19 preorders

i.e. together 29 preorders.

• for n=4:

• 1 partition of 4, giving 1 preorder• 7 partitions with two classes (4 of 3+1 and 3 of 2+2), giving 7 × 3 = 21 preorders• 6 partitions of 2+1+1, giving 6 × 19 = 114 preorders

4.6. INTERVAL 37

• 1 partition of 1+1+1+1, giving 219 preorders

i.e. together 355 preorders.

4.6 Interval

For a ≲ b, the interval [a,b] is the set of points x satisfying a ≲ x and x ≲ b, also written a ≲ x ≲ b. It contains atleast the points a and b. One may choose to extend the definition to all pairs (a,b). The extra intervals are all empty.Using the corresponding strict relation "<", one can also define the interval (a,b) as the set of points x satisfying a <x and x < b, also written a < x < b. An open interval may be empty even if a < b.Also [a,b) and (a,b] can be defined similarly.

4.7 See also• partial order - preorder that is antisymmetric

• equivalence relation - preorder that is symmetric

• total preorder - preorder that is total

• total order - preorder that is antisymmetric and total

• directed set

• category of preordered sets

• prewellordering

• Well-quasi-ordering

4.8 References[1] For “proset”, see e.g. Eklund, Patrik; Gähler, Werner (1990), “Generalized Cauchy spaces”, Mathematische Nachrichten

147: 219–233, doi:10.1002/mana.19901470123, MR 1127325.

• Schröder, Bernd S. W. (2002), Ordered Sets: An Introduction, Boston: Birkhäuser, ISBN 0-8176-4128-9

Chapter 5

Prewellordering

In set theory, a prewellordering is a binary relation ≤ that is transitive, total, and wellfounded (more precisely, therelation x ≤ y ∧ y ≰ x is wellfounded). In other words, if ≤ is a prewellordering on a set X , and if we define ∼ by

x ∼ y ⇐⇒ x ≤ y ∧ y ≤ x

then ∼ is an equivalence relation on X , and ≤ induces a wellordering on the quotient X/ ∼ . The order-type of thisinduced wellordering is an ordinal, referred to as the length of the prewellordering.A norm on a set X is a map from X into the ordinals. Every norm induces a prewellordering; if ϕ : X → Ord is anorm, the associated prewellordering is given by

x ≤ y ⇐⇒ ϕ(x) ≤ ϕ(y)

Conversely, every prewellordering is induced by a unique regular norm (a norm ϕ : X → Ord is regular if, for anyx ∈ X and any α < ϕ(x) , there is y ∈ X such that ϕ(y) = α ).

5.1 Prewellordering property

If Γ is a pointclass of subsets of some collection F of Polish spaces, F closed under Cartesian product, and if ≤ is aprewellordering of some subset P of some element X of F , then ≤ is said to be a Γ -prewellordering of P if therelations <∗ and ≤∗ are elements of Γ , where for x, y ∈ X ,

1. x <∗ y ⇐⇒ x ∈ P ∧ [y /∈ P ∨ {x ≤ y ∧ y ̸≤ x}]

2. x ≤∗ y ⇐⇒ x ∈ P ∧ [y /∈ P ∨ x ≤ y]

Γ is said to have the prewellordering property if every set in Γ admits a Γ -prewellordering.The prewellordering property is related to the stronger scale property; in practice, many pointclasses having theprewellordering property also have the scale property, which allows drawing stronger conclusions.

5.1.1 Examples

Π11 andΣ1

2 both have the prewellordering property; this is provable in ZFC alone. Assuming sufficient large cardinals,for every n ∈ ω , Π1

2n+1 and Σ12n+2 have the prewellordering property.

5.1.2 Consequences

38

5.2. SEE ALSO 39

Reduction

If Γ is an adequate pointclass with the prewellordering property, then it also has the reduction property: For anyspace X ∈ F and any sets A,B ⊆ X , A and B both in Γ , the union A ∪B may be partitioned into sets A∗, B∗ ,both in Γ , such that A∗ ⊆ A and B∗ ⊆ B .

Separation

If Γ is an adequate pointclass whose dual pointclass has the prewellordering property, then Γ has the separationproperty: For any space X ∈ F and any sets A,B ⊆ X , A and B disjoint sets both in Γ , there is a set C ⊆ Xsuch that both C and its complement X \ C are in Γ , with A ⊆ C and B ∩ C = ∅ .For example, Π1

1 has the prewellordering property, so Σ11 has the separation property. This means that if A and B

are disjoint analytic subsets of some Polish space X , then there is a Borel subset C of X such that C includes A andis disjoint from B .

5.2 See also• Descriptive set theory

• Scale property

• Graded poset – a graded poset is analogous to a prewellordering with a norm, replacing a map to the ordinalswith a map to the integers

5.3 References• Moschovakis, Yiannis N. (1980). Descriptive Set Theory. North Holland. ISBN 0-444-70199-0.

Chapter 6

Quasitransitive relation

Quasitransitivity is a weakened version of transitivity that is used in social choice theory or microeconomics. In-formally, a relation is quasitransitive if it is symmetric for some values and transitive elsewhere. The concept wasintroduced by Sen (1969) to study the consequences of Arrow’s theorem.

6.1 Formal definition

A binary relation T over a set X is quasitransitive if for all a, b, and c in X the following holds:

(aT b) ∧ ¬(bT a) ∧ (bT c) ∧ ¬(cT b) ⇒ (aT c) ∧ ¬(cT a).

If the relation is also antisymmetric, T is transitive.Alternately, for a relation T, define the asymmetric or “strict” part P:

(aP b) ⇔ (aT b) ∧ ¬(bT a).

Then T is quasitransitive iff P is transitive.

6.2 Examples

Preferences are assumed to be quasitransitive (rather than transitive) in some economic contexts. The classic exampleis a person indifferent between 10 and 11 grams of sugar and indifferent between 11 and 12 grams of sugar, but whoprefers 12 grams of sugar to 10. Similarly, the Sorites paradox can be resolved by weakening assumed transitivity ofcertain relations to quasitransitivity.

6.3 Properties

• Every transitive relation is quasitransitive; every quasitransitive relation is an acyclic relation. In each case theconverse does not hold in general.

6.4 See also

• Intransitivity

• Reflexive relation

40

6.5. REFERENCES 41

6.5 References• Bossert, Walter; Suzumura, Kōtarō (2010). Consistency, choice and rationality. Harvard University Press.

ISBN 0674052994.

• Sen, A. (1969). “Quasi-transitivity, rational choice and collective decisions”. Rev. Econ. Stud. 36: 381–393.doi:10.2307/2296434. Zbl 0181.47302.

Chapter 7

Quotient by an equivalence relation

This article is about a generalization to category theory, used in scheme theory. For the common meaning, seeEquivalence class.

In mathematics, given a category C, a quotient of an object X by an equivalence relation f : R → X × X is acoequalizer for the pair of maps

Rf→X ×X

pri→X, i = 1, 2,

where R is an object in C and "f is an equivalence relation” means that, for any object T in C, the image (which isa set) of f : R(T ) = Mor(T,R) → X(T ) × X(T ) is an equivalence relation; that is, (x, y) is in it if and only if(y, x) is in it, etc.The basic case in practice is when C is the category of all schemes over some scheme S. But the notion is flexible andone can also take C to be the category of sheaves.

7.1 Examples

• Let X be a set and consider some equivalence relation on it. Let Q be the set of all equivalence classes in X.Then the map q : X → Q that sends an element x to an equivalence class to which x belong is a quotient.

• In the above example, Q is a subset of the power set H of X. In algebraic geometry, one might replace Hby a Hilbert scheme or disjoint union of Hilbert schemes. In fact, Grothendieck constructed a relative Picardscheme of a flat projective schemeX[1] as a quotientQ (of the scheme Z parametrizing relative effective divisorson X) that is a closed scheme of a Hilbert scheme H. The quotient map q : Z → Q can then be thought of asa relative version of the Abel map.

7.2 See also

• categorical quotient, a special case

7.3 Notes

[1] One also needs to assume the geometric fibers are integral schemes; Mumford’s example shows the “integral” cannot beomitted.

42

7.4. REFERENCES 43

7.4 References• Nitsure, N. Construction of Hilbert and Quot schemes. Fundamental algebraic geometry: Grothendieck’s FGA

explained, Mathematical Surveys and Monographs 123, American Mathematical Society 2005, 105–137.

Chapter 8

Reduct

This article is about a relation on algebraic structures. For reducts in abstract rewriting, see Confluence (abstractrewriting).

In universal algebra and in model theory, a reduct of an algebraic structure is obtained by omitting some of theoperations and relations of that structure. The converse of “reduct” is “expansion.”

8.1 Definition

Let A be an algebraic structure (in the sense of universal algebra) or equivalently a structure in the sense of modeltheory, organized as a set X together with an indexed family of operations and relations φᵢ on that set, with index setI. Then the reduct of A defined by a subset J of I is the structure consisting of the set X and J-indexed family ofoperations and relations whose j-th operation or relation for j∈J is the j-th operation or relation of A. That is, thisreduct is the structure A with the omission of those operations and relations φi for which i is not in J.A structure A is an expansion of B just when B is a reduct of A. That is, reduct and expansion are mutual converses.

8.2 Examples

The monoid (Z, +, 0) of integers under addition is a reduct of the group (Z, +, −, 0) of integers under addition andnegation, obtained by omitting negation. By contrast, the monoid (N,+,0) of natural numbers under addition is notthe reduct of any group.Conversely the group (Z, +, −, 0) is the expansion of the monoid (Z, +, 0), expanding it with the operation of negation.

8.3 References• Burris, Stanley N.; H. P. Sankappanavar (1981). ACourse in Universal Algebra. Springer. ISBN 3-540-90578-

2.

• Hodges, Wilfrid (1993). Model theory. Cambridge University Press. ISBN 0-521-30442-3.

44

Chapter 9

Semiorder

In order theory, a branch of mathematics, a semiorder is a type of ordering that may be determined for a set ofitems with numerical scores by declaring two items to be incomparable when their scores are within a given marginof error of each other, and by using the numerical comparison of their scores when those scores are sufficiently farapart. Semiorders were introduced and applied in mathematical psychology by Luce (1956) as a model of humanpreference without the assumption that indifference is transitive. They generalize strict weak orderings, form a specialcase of partial orders and interval orders, and can be characterized among the partial orders by two forbidden four-item suborders.

9.1 Definition

Let X be a set of items, and let < be a binary relation on X. Items x and y are said to be incomparable, written here asx ~ y, if neither x < y nor y < x is true. Then the pair (X,<) is a semiorder if it satisfies the following three axioms:[1]

• For all x and y, it is not possible for both x < y and y < x to be true. That is, < must be an irreflexive,antisymmetric relation

• For all x, y, z, and w, if it is true that x < y, y ~ z, and z < w, then it must also be true that x < w.

• For all x, y, z, and w, if it is true that x < y, y < z, and y ~ w, then it cannot also be true that x ~ w and z ~ wsimultaneously.

It follows from the first axiom that x ~ x, and therefore the second axiom (with y = z) implies that < is a transitiverelation.One may define a partial order (X,≤) from a semiorder (X,<) by declaring that x ≤ ywhenever either x < y or x = y. Ofthe axioms that a partial order is required to obey, reflexivity follows automatically from this definition, antisymmetryfollows from the first semiorder axiom, and transitivity follows from the second semiorder axiom. Conversely, froma partial order defined in this way, the semiorder may be recovered by declaring that x < y whenever x ≤ y and x ≠y. The first of the semiorder axioms listed above follows automatically from the axioms defining a partial order, butthe others do not. The second and third semiorder axioms forbid partial orders of four items forming two disjointchains: the second axiom forbids two chains of two items each, while the third item forbids a three-item chain withone unrelated item.

9.2 Utility

The original motivation for introducing semiorders was to model human preferences without assuming (as strict weakorderings do) that incomparability is a transitive relation. For instance, if x, y, and z represent three quantities of thesame material, and x and z differ by the smallest amount that is perceptible as a difference, while y is halfway betweenthe two of them, then it is reasonable for a preference to exist between x and z but not between the other two pairs,violating transitivity.[2]

45

46 CHAPTER 9. SEMIORDER

An example of a semiorder, shown by its Hasse diagram. The horizontal blue lines indicate the spacing of the y-coordinates of thepoints; two points are comparable when their y coordinates differ by at least one unit.

Thus, suppose that X is a set of items, and u is a utility function that maps the members of X to real numbers. A strictweak ordering can be defined on x by declaring two items to be incomparable when they have equal utilities, andotherwise using the numerical comparison, but this necessarily leads to a transitive incomparability relation. Instead,if one sets a numerical threshold (which may be normalized to 1) such that utilities within that threshold of each otherare declared incomparable, then a semiorder arises.Specifically, define a binary relation < from X and u by setting x < y whenever u(x) ≤ u(y) − 1. Then (X,<) is asemiorder.[3] It may equivalently be defined as the interval order defined by the intervals [u(x),u(x) + 1].[4]

The converse is not necessarily true: for instance, if a semiorder (X,<) includes an uncountable totally ordered subsetthen there do not exist sufficiently many sufficiently well-spaced real-numbers to represent this subset numerically.However, every finite semiorder can be defined from a utility function in this way.[5] Fishburn (1973) supplies aprecise characterization of the semiorders that may be defined numerically.

9.3. OTHER RESULTS 47

9.3 Other results

The number of distinct semiorders on n unlabeled items is given by the Catalan numbers

1n+1

(2nn

), [6]

while the number of semiorders on n labeled items is given by the sequence

1, 1, 3, 19, 183, 2371, 38703, 763099, 17648823, ... (sequence A006531 in OEIS).[7]

Any finite semiorder has order dimension at most three.[8]

Among all partial orders with a fixed number of elements and a fixed number of comparable pairs, the partial ordersthat have the largest number of linear extensions are semiorders.[9]

Semiorders are known to obey the 1/3–2/3 conjecture: in any finite semiorder that is not a total order, there existsa pair of elements x and y such that x appears earlier than y in between 1/3 and 2/3 of the linear extensions of thesemiorder.[10]

The set of semiorders on an n-element set is well-graded: if two semiorders on the same set differ from each otherby the addition or removal of k order relations, then it is possible to find a path of k steps from the first semiorder tothe second one, in such a way that each step of the path adds or removes a single order relation and each intermediatestate in the path is itself a semiorder.[11]

The incomparability graphs of semiorders are called indifference graphs, and are a special case of the intervalgraphs.[12]

9.4 Notes

[1] Luce (1956) describes an equivalent set of four axioms, the first two of which combine the definition of incomparabilityand the first axiom listed here.

[2] Luce (1956), p. 179.

[3] Luce (1956), Theorem 3 describes a more general situation in which the threshold for comparability between two utilitiesis a function of the utility rather than being identically 1.

[4] Fishburn (1970).

[5] This result is typically credited to Scott & Suppes (1958); see, e.g., Rabinovitch (1977). However, Luce (1956), Theorem2 proves a more general statement, that a finite semiorder can be defined from a utility function and a threshold functionwhenever a certain underlying weak order can be defined numerically. For finite semiorders, it is trivial that the weak ordercan be defined numerically with a unit threshold function.

[6] Kim & Roush (1978).

[7] Chandon, Lemaire & Pouget (1978).

[8] Rabinovitch (1978).

[9] Fishburn & Trotter (1992).

[10] Brightwell (1989).

[11] Doignon & Falmagne (1997).

[12] Roberts, Fred S. (1969), “Indifference graphs”, Proof Techniques in Graph Theory (Proc. Second Ann Arbor Graph TheoryConf., Ann Arbor, Mich., 1968), Academic Press, New York, pp. 139–146, MR 0252267.

48 CHAPTER 9. SEMIORDER

9.5 References• Brightwell, Graham R. (1989), “Semiorders and the 1/3–2/3 conjecture”,Order 5 (4): 369–380, doi:10.1007/BF00353656.

• Chandon, J.-L.; Lemaire, J.; Pouget, J. (1978), “Dénombrement des quasi-ordres sur un ensemble fini”, Centrede Mathématique Sociale. École Pratique des Hautes Études. Mathématiques et Sciences Humaines (62): 61–80,83, MR 517680.

• Doignon, Jean-Paul; Falmagne, Jean-Claude (1997), “Well-graded families of relations”, Discrete Mathematics173 (1-3): 35–44, doi:10.1016/S0012-365X(96)00095-7, MR 1468838.

• Fishburn, Peter C. (1970), “Intransitive indifference with unequal indifference intervals”, J. MathematicalPsychology 7: 144–149, doi:10.1016/0022-2496(70)90062-3, MR 0253942.

• Fishburn, Peter C. (1973), “Interval representations for interval orders and semiorders”, J. Mathematical Psy-chology 10: 91–105, doi:10.1016/0022-2496(73)90007-2, MR 0316322.

• Fishburn, Peter C.; Trotter, W. T. (1992), “Linear extensions of semiorders: a maximization problem”,DiscreteMathematics 103 (1): 25–40, doi:10.1016/0012-365X(92)90036-F, MR 1171114.

• Kim, K. H.; Roush, F. W. (1978), “Enumeration of isomorphism classes of semiorders”, Journal of Combina-torics, Information &System Sciences 3 (2): 58–61, MR 538212.

• Luce, R. Duncan (1956), “Semiorders and a theory of utility discrimination”, Econometrica 24: 178–191,JSTOR 1905751, MR 0078632.

• Rabinovitch, Issie (1977), “The Scott-Suppes theorem on semiorders”, J. Mathematical Psychology 15 (2):209–212, doi:10.1016/0022-2496(77)90030-x, MR 0437404.

• Rabinovitch, Issie (1978), “The dimension of semiorders”, Journal of Combinatorial Theory. Series A 25 (1):50–61, doi:10.1016/0097-3165(78)90030-4, MR 0498294.

• Scott, Dana; Suppes, Patrick (1958), “Foundational aspects of theories of measurement”, The Journal of Sym-bolic Logic 23: 113–128, doi:10.2307/2964389, MR 0115919.

9.6 Additional reading• Pirlot, M.; Vincke, Ph. (1997), Semiorders: Properties, representations, applications, Theory and Decision

Library. Series B: Mathematical and Statistical Methods 36, Dordrecht: Kluwer Academic Publishers Group,ISBN 0-7923-4617-3, MR 1472236.

Chapter 10

Separoid

In mathematics, a separoid is a binary relation between disjoint sets which is stable as an ideal in the canonical orderinduced by inclusion. Many mathematical objects which appear to be quite different, find a common generalisation inthe framework of separoids; e.g., graphs, configurations of convex sets, oriented matroids, and polytopes. Any count-able category is an induced subcategory of separoids when they are endowed with homomorphisms (viz., mappingsthat preserve the so-called minimal Radon partitions).In this general framework, some results and invariants of different categories turn out to be special cases of thesame aspect; e.g., the pseudoachromatic number from graph theory and the Tverberg theorem from combinatorialconvexity are simply two faces of the same aspect, namely, complete colouring of separoids.

10.1 The axioms

A separoid is a set S endowed with a binary relation | ⊆ 2S × 2S on its power set, which satisfies the followingsimple properties for A,B ⊆ S :

A | B ⇔ B | A,

A | B ⇒ A ∩B = ∅,

A | B and A′ ⊂ A ⇒ A′ | B.

A related pair A | B is called a separation and we often say that A is separated from B. It is enough to know themaximal separations to reconstruct the separoid.A mapping φ : S → T is a morphism of separoids if the preimages of separations are separations; that is, forA,B ⊆ T

A | B ⇒ φ−1(A) | φ−1(B).

10.2 Examples

Examples of separoids can be found in almost every branch of mathematics. Here we list just a few.1. Given a graph G=(V,E), we can define a separoid on its vertices by saying that two (disjoint) subsets of V, say Aand B, are separated if there are no edges going from one to the other; i.e.,

A | B ⇔ ∀a ∈ A and b ∈ B : ab ̸∈ E.

49

50 CHAPTER 10. SEPAROID

2. Given an oriented matroid M = (E,T), given in terms of its topes T, we can define a separoid on E by saying thattwo subsets are separated if they are contained in opposite signs of a tope. In other words, the topes of an orientedmatroid are the maximal separations of a separoid. This example includes, of course, all directed graphs.3. Given a family of objects in an Euclidean space, we can define a separoid in it by saying that two subsets areseparated if there exists a hyperplane that separates them; i.e., leaving them in the two opposite sides of it.4. Given a topological space, we can define a separoid saying that two subsets are separated if there exist two disjointopen sets which contains them (one for each of them).

10.3 The basic lemma

Every separoid can be represented with a family of convex sets in some Euclidean space and their separations byhyperplanes.

10.4 References• Strausz Ricardo; “Separoides”. Situs, serie B, no. 5 (1998), Universidad Nacional Autónoma de México.

• Arocha Jorge Luis, Bracho Javier, Montejano Luis, Oliveros Deborah, Strausz Ricardo; “Separoids, their cat-egories and a Hadwiger-type theorem for transversals”. Discrete and Computational Geometry 27 (2002), no.3, 377–385.

• Strausz Ricardo; “Separoids and a Tverberg-type problem”. Geombinatorics 15 (2005), no. 2, 79–92.

• Montellano-Ballesteros Juan Jose, Por Attila, Strausz Ricardo; “Tverberg-type theorems for separoids”. Dis-crete and Computational Geometry 35 (2006), no.3, 513–523.

• Nešetřil Jaroslav, Strausz Ricardo; “Universality of separoids”. ArchivumMathematicum (Brno) 42 (2006), no.1, 85–101.

• Bracho Javier, Strausz Ricardo; “Two geometric representations of separoids”. Periodica Mathematica Hun-garica 53 (2006), no. 1-2, 115–120.

• Strausz Ricardo; “Homomorphisms of separoids”. 6th Czech-Slovak International Symposium on Combina-torics, Graph Theory, Algorithms and Applications, 461–468, Electronic Notes on Discrete Mathematics 28,Elsevier, Amsterdam, 2007.

• Strausz Ricardo; “Edrös-Szekeres 'happy end'-type theorems for separoids”. European Journal of Combina-torics 29 (2008), no. 4, 1076–1085.

Chapter 11

Series-parallel partial order

Series composition

Parallel composition

A series-parallel partial order, shown as a Hasse diagram.

In order-theoretic mathematics, a series-parallel partial order is a partially ordered set built up from smaller series-

51

52 CHAPTER 11. SERIES-PARALLEL PARTIAL ORDER

parallel partial orders by two simple composition operations.[1][2]

The series-parallel partial orders may be characterized as the N-free finite partial orders; they have order dimension atmost two.[1][3] They include weak orders and the reachability relationship in directed trees and directed series-parallelgraphs.[2][3] The comparability graphs of series-parallel partial orders are cographs.[2][4]

Series-parallel partial orders have been applied in job shop scheduling,[5] machine learning of event sequencing intime series data,[6] transmission sequencing of multimedia data,[7] and throughput maximization in dataflow pro-gramming.[8]

Series-parallel partial orders have also been called multitrees;[4] however, that name is ambiguous: multitrees alsorefer to partial orders with no four-element diamond suborder[9] and to other structures formed from multiple trees.

11.1 Definition

Consider P and Q, two partially ordered sets. The series composition of P and Q, written P; Q,[7] P * Q,[2] or P ⧀Q,[1]is the partially ordered set whose elements are the disjoint union of the elements of P andQ. In P;Q, two elementsx and y that both belong to P or that both belong to Q have the same order relation that they do in P or Q respectively.However, for every pair x, y where x belongs to P and y belongs to Q, there is an additional order relation x ≤ y in theseries composition. Series composition is an associative operation: one can write P; Q; R as the series compositionof three orders, without ambiguity about how to combine them pairwise, because both of the parenthesizations (P;Q); R and P; (Q; R) describe the same partial order. However, it is not a commutative operation, because switchingthe roles of P and Q will produce a different partial order that reverses the order relations of pairs with one elementin P and one in Q.[1]

The parallel composition of P and Q, written P || Q,[7] P + Q,[2] or P ⊕ Q,[1] is defined similarly, from the disjointunion of the elements in P and the elements in Q, with pairs of elements that both belong to P or both to Q havingthe same order as they do in P or Q respectively. In P || Q, a pair x, y is incomparable whenever x belongs to P and ybelongs to Q. Parallel composition is both commutative and associative.[1]

The class of series-parallel partial orders is the set of partial orders that can be built up from single-element partialorders using these two operations. Equivalently, it is the smallest set of partial orders that includes the single-elementpartial order and is closed under the series and parallel composition operations.[1][2]

A weak order is the series parallel partial order obtained from a sequence of composition operations in which allof the parallel compositions are performed first, and then the results of these compositions are combined using onlyseries compositions.[2]

11.2 Forbidden suborder characterization

The partial order N with the four elements a, b, c, and d and exactly the three order relations a ≤ b ≥ c ≤ d is anexample of a fence or zigzag poset; its Hasse diagram has the shape of the capital letter “N”. It is not series-parallel,because there is no way of splitting it into the series or parallel composition of two smaller partial orders. A partialorder P is said to be N-free if there does not exist a set of four elements in P such that the restriction of P to thoseelements is order-isomorphic to N. The series-parallel partial orders are exactly the nonempty finite N-free partialorders.[1][2][3]

It follows immediately from this (although it can also be proven directly) that any nonempty restriction of a series-parallel partial order is itself a series-parallel partial order.[1]

11.3 Order dimension

The order dimension of a partial order P is the minimum size of a realizer of P, a set of linear extensions of P withthe property that, for every two distinct elements x and y of P, x ≤ y in P if and only if x has an earlier position thany in every linear extension of the realizer. Series-parallel partial orders have order dimension at most two. If P andQ have realizers {L1, L2} and {L3, L4}, respectively, then {L1L3, L2L4} is a realizer of the series composition P; Q,and {L1L3, L4L2} is a realizer of the parallel composition P || Q.[2][3] A partial order is series-parallel if and only ifit has a realizer in which one of the two permutations is the identity and the other is a separable permutation.

11.4. CONNECTIONS TO GRAPH THEORY 53

It is known that a partial order P has order dimension two if and only if there exists a conjugate order Q on thesame elements, with the property that any two distinct elements x and y are comparable on exactly one of these twoorders. In the case of series parallel partial orders, a conjugate order that is itself series parallel may be obtained byperforming a sequence of composition operations in the same order as the ones defining P on the same elements,but performing a series composition for each parallel composition in the decomposition of P and vice versa. Morestrongly, although a partial order may have many different conjugates, every conjugate of a series parallel partial ordermust itself be series parallel.[2]

11.4 Connections to graph theory

Any partial order may be represented (usually in more than one way) by a directed acyclic graph in which there is apath from x to ywhenever x and y are elements of the partial order with x ≤ y. The graphs that represent series-parallelpartial orders in this way have been called vertex series parallel graphs, and their transitive reductions (the graphsof the covering relations of the partial order) are called minimal vertex series parallel graphs.[3] Directed trees and(two-terminal) series parallel graphs are examples of minimal vertex series parallel graphs; therefore, series parallelpartial orders may be used to represent reachability relations in directed trees and series parallel graphs.[2][3]

The comparability graph of a partial order is the undirected graph with a vertex for each element and an undirectededge for each pair of distinct elements x, y with either x ≤ y or y ≤ x. That is, it is formed from a minimal vertexseries parallel graph by forgetting the orientation of each edge. The comparability graph of a series-partial order is acograph: the series and parallel composition operations of the partial order give rise to operations on the comparabilitygraph that form the disjoint union of two subgraphs or that connect two subgraphs by all possible edges; these twooperations are the basic operations from which cographs are defined. Conversely, every cograph is the comparabilitygraph of a series-parallel partial order. If a partial order has a cograph as its comparability graph, then it must be aseries-parallel partial order, because every other kind of partial order has an N suborder that would correspond to aninduced four-vertex path in its comparability graph, and such paths are forbidden in cographs.[2][4]

11.5 Computational complexity

It is possible to use the forbidden suborder characterization of series-parallel partial orders as a basis for an algorithmthat tests whether a given binary relation is a series-parallel partial order, in an amount of time that is linear inthe number of related pairs.[2][3] Alternatively, if a partial order is described as the reachability order of a directedacyclic graph, it is possible to test whether it is a series-parallel partial order, and if so compute its transitive closure,in time proportional to the number of vertices and edges in the transitive closure; it remains open whether the timeto recognize series-parallel reachability orders can be improved to be linear in the size of the input graph.[10]

If a series-parallel partial order is represented as an expression tree describing the series and parallel compositionoperations that formed it, then the elements of the partial order may be represented by the leaves of the expressiontree. A comparison between any two elements may be performed algorithmically by searching for the lowest commonancestor of the corresponding two leaves; if that ancestor is a parallel composition, the two elements are incomparable,and otherwise the order of the series composition operands determines the order of the elements. In this way, a series-parallel partial order on n elements may be represented in O(n) space with O(1) time to determine any comparisonvalue.[2]

It is NP-complete to test, for two given series-parallel partial orders P and Q, whether P contains a restriction iso-morphic to Q.[3]

Although the problem of counting the number of linear extensions of an arbitrary partial order is #P-complete,[11] itmay be solved in polynomial time for series-parallel partial orders. Specifically, if L(P) denotes the number of linearextensions of a partial order P, then L(P; Q) = L(P)L(Q) and

L(P ||Q) =(|P |+ |Q|)!|P |!|Q|!

L(P )L(Q),

so the number of linear extensions may be calculated using an expression tree with the same form as the decompositiontree of the given series-parallel order.[2]

54 CHAPTER 11. SERIES-PARALLEL PARTIAL ORDER

11.6 Applications

Mannila & Meek (2000) use series-parallel partial orders as a model for the sequences of events in time series data.They describe machine learning algorithms for inferring models of this type, and demonstrate its effectiveness atinferring course prerequisites from student enrollment data and at modeling web browser usage patterns.[6]

Amer et al. (1994) argue that series-parallel partial orders are a good fit for modeling the transmission sequencingrequirements of multimedia presentations. They use the formula for computing the number of linear extensions of aseries-parallel partial order as the basis for analyzing multimedia transmission algorithms.[7]

Choudhary et al. (1994) use series-parallel partial orders to model the task dependencies in a dataflow model ofmassive data processing for computer vision. They show that, by using series-parallel orders for this problem, it ispossible to efficiently construct an optimized schedule that assigns different tasks to different processors of a parallelcomputing system in order to optimize the throughput of the system.[8]

A class of orderings somewhat more general than series-parallel partial orders is provided by PQ trees, data structuresthat have been applied in algorithms for testing whether a graph is planar and recognizing interval graphs.[12] A Pnode of a PQ tree allows all possible orderings of its children, like a parallel composition of partial orders, while a Qnode requires the children to occur in a fixed linear ordering, like a series composition of partial orders. However,unlike series-parallel partial orders, PQ trees allow the linear ordering of any Q node to be reversed.

11.7 See also• Series and parallel circuits

11.8 References[1] Bechet, Denis; De Groote, Philippe; Retoré, Christian (1997), “A complete axiomatisation for the inclusion of series-

parallel partial orders”, Rewriting Techniques and Applications, Lecture Notes in Computer Science 1232, Springer-Verlag,pp. 230–240, doi:10.1007/3-540-62950-5_74.

[2] Möhring, Rolf H. (1989), “Computationally tractable classes of ordered sets”, in Rival, Ivan, Algorithms and Order: Pro-ceedings of the NATO Advanced Study Institute on Algorithms and Order, Ottawa, Canada, May 31-June 13, 1987, NATOScience Series C 255, Springer-Verlag, pp. 105–194, ISBN 978-0-7923-0007-6.

[3] Valdes, Jacobo; Tarjan, Robert E.; Lawler, Eugene L. (1982), “The recognition of series parallel digraphs”, SIAM Journalon Computing 11 (2): 298–313, doi:10.1137/0211023.

[4] Jung, H. A. (1978), “On a class of posets and the corresponding comparability graphs”, Journal of Combinatorial Theory,Series B 24 (2): 125–133, doi:10.1016/0095-8956(78)90013-8, MR 0491356.

[5] Lawler, Eugene L. (1978), “Sequencing jobs to minimize total weighted completion time subject to precedence constraints”,Annals of Discrete Mathematics 2: 75–90, doi:10.1016/S0167-5060(08)70323-6, MR 0495156.

[6] Mannila, Heikki; Meek, Christopher (2000), “Global partial orders from sequential data”, Proc. 6th ACM SIGKDD Inter-national Conference on Knowledge Discovery and Data Mining (KDD 2000), pp. 161–168, doi:10.1145/347090.347122.

[7] Amer, Paul D.; Chassot, Christophe; Connolly, Thomas J.; Diaz, Michel; Conrad, Phillip (1994), “Partial-order transportservice for multimedia and other applications”, IEEE/ACMTransactions onNetworking 2 (5): 440–456, doi:10.1109/90.336326.

[8] Choudhary, A. N.; Narahari, B.; Nicol, D. M.; Simha, R. (1994), “Optimal processor assignment for a class of pipelinedcomputations”, IEEE Transactions on Parallel and Distributed Systems 5 (4): 439–445, doi:10.1109/71.273050.

[9] Furnas, George W.; Zacks, Jeff (1994), “Multitrees: enriching and reusing hierarchical structure”, Proc. SIGCHI conferenceon Human Factors in Computing Systems (CHI '94), pp. 330–336, doi:10.1145/191666.191778.

[10] Ma, Tze-Heng; Spinrad, Jeremy (1991), “Transitive closure for restricted classes of partial orders”, Order 8 (2): 175–183,doi:10.1007/BF00383402.

[11] Brightwell, Graham R.; Winkler, Peter (1991), “Counting linear extensions”,Order 8 (3): 225–242, doi:10.1007/BF00383444.

[12] Booth, Kellogg S.; Lueker, George S. (1976), “Testing for the consecutive ones property, interval graphs, and graphplanarity using PQ-tree algorithms”, Journal of Computer and System Sciences 13 (3): 335–379, doi:10.1016/S0022-0000(76)80045-1.

Chapter 12

Weak ordering

Not to be confused with weak order of permutations.In mathematics, especially order theory, a weak ordering is a mathematical formalization of the intuitive notion of

a<bc<b

a<ba<c

b<ac<a

b<ab<c

c<ac<b

a<cb<ca,b,c

c<b<a

b<c<a

b<a<c

c<a<b a<b<c

a<c<b

The 13 possible strict weak orderings on a set of three elements {a, b, c}. The only partially ordered sets are coloured, while totallyordered ones are in black. Two orderings are shown as connected by an edge if they differ by a single dichotomy.

55

56 CHAPTER 12. WEAK ORDERING

a ranking of a set, some of whose members may be tied with each other. Weak orders are a generalization of totallyordered sets (rankings without ties) and are in turn generalized by partially ordered sets and preorders.[1]

There are several common ways of formalizing weak orderings, that are different from each other but cryptomorphic(interconvertable with no loss of information): they may be axiomatized as strict weak orderings (partially orderedsets in which incomparability is a transitive relation), as total preorders (transitive binary relations in which at leastone of the two possible relations exists between every pair of elements), or as ordered partitions (partitions of theelements into disjoint subsets, together with a total order on the subsets). In many cases another representation calleda preferential arrangement based on a utility function is also possible.Weak orderings are counted by the ordered Bell numbers. They are used in computer science as part of partitionrefinement algorithms, and in the C++ Standard Library.

12.1 Examples

In horse racing, the use of photo finishes has eliminated some, but not all, ties or (as they are called in this context)dead heats, so the outcome of a horse race may be modeled by a weak ordering.[2] In an example from the MarylandHunt Cup steeplechase in 2007, The Bruce was the clear winner, but two horses Bug River and Lear Charm tied forsecond place, with the remaining horses farther back; three horses did not finish.[3] In the weak ordering describingthis outcome, The Bruce would be first, Bug River and Lear Charm would be ranked after The Bruce but before allthe other horses that finished, and the three horses that did not finish would be placed last in the order but tied witheach other.The points of the Euclidean plane may be ordered by their distance from the origin, giving another example of a weakordering with infinitely many elements, infinitely many subsets of tied elements (the sets of points that belong to acommon circle centered at the origin), and infinitely many points within these subsets. Although this ordering has asmallest element (the origin itself), it does not have any second-smallest elements, nor any largest element.Opinion polling in political elections provides an example of a type of ordering that resembles weak orderings, but isbetter modeled mathematically in other ways. In the results of a poll, one candidate may be clearly ahead of another,or the two candidates may be statistically tied, meaning not that their poll results are equal but rather that they arewithin the margin of error of each other. However, if candidate x is statistically tied with y, and y is statistically tiedwith z, it might still be possible for x to be clearly better than z, so being tied is not in this case a transitive relation.Because of this possibility, rankings of this type are better modeled as semiorders than as weak orderings.[4]

12.2 Axiomatizations

12.2.1 Strict weak orderings

A strict weak ordering is a binary relation < on a set S that is a strict partial order (a transitive relation that isirreflexive, or equivalently,[5] that is asymmetric) in which the relation “neither a < b nor b < a" is transitive.[1]

The equivalence classes of this “incomparability relation” partition the elements of S, and are totally ordered by <.Conversely, any total order on a partition of S gives rise to a strict weak ordering in which x < y if and only if thereexists sets A and B in the partition with x in A, y in B, and A < B in the total order.As a non-example, consider the partial order in the set {a, b, c} defined by the relationship b < c. The pairs a,b and a,care incomparable but b and c are related, so incomparability does not form an equivalence relation and this exampleis not a strict weak ordering.A strict weak ordering has the following properties. For all x and y in S,

• For all x, it is not the case that x < x (irreflexivity).

• For all x, y, if x < y then it is not the case that y < x (asymmetry).

• For all x, y, and z, if x < y and y < z then x < z (transitivity).

• For all x, y, and z, if x is incomparable with y, and y is incomparable with z, then x is incomparable with z(transitivity of incomparability).

12.2. AXIOMATIZATIONS 57

This list of properties is somewhat redundant, as asymmetry follows readily from irreflexivity and transitivity.Transitivity of incomparability (together with transitivity) can also be stated in the following forms:

• If x < y, then for all z, either x < z or z < y or both.

Or:

• If x is incomparable with y, then for all z ≠ x, z ≠ y, either (x < z and y < z) or (z < x and z < y) or (z isincomparable with x and z is incomparable with y).

12.2.2 Total preorders

Strict weak orders are very closely related to total preorders or (non-strict) weak orders, and the same mathematicalconcepts that can be modeled with strict weak orderings can be modeled equally well with total preorders. A totalpreorder or weak order is a preorder that is total; that is, no pair of items is incomparable. A total preorder ≲ satisfiesthe following properties:

• For all x, y, and z, if x ≲ y and y ≲ z then x ≲ z (transitivity).

• For all x and y, x ≲ y or y ≲ x (totality).

• Hence, for all x, x ≲ x (reflexivity).

A total order is a total preorder which is antisymmetric, in other words, which is also a partial order. Total preordersare sometimes also called preference relations.The complement of a strict weak order is a total preorder, and vice versa, but it seems more natural to relate strictweak orders and total preorders in a way that preserves rather than reverses the order of the elements. Thus we takethe inverse of the complement: for a strict weak ordering <, define a total preorder ≲ by setting x ≲ y whenever it isnot the case that y < x. In the other direction, to define a strict weak ordering < from a total preorder ≲ , set x < ywhenever it is not the case that y ≲ x.[6]

In any preorder there is a corresponding equivalence relation where two elements x and y are defined as equivalent ifx ≲ y and y ≲ x. In the case of a total preorder the corresponding partial order on the set of equivalence classes is atotal order. Two elements are equivalent in a total preorder if and only if they are incomparable in the correspondingstrict weak ordering.

12.2.3 Ordered partitions

A partition of a set S is a family of disjoint subsets of S that have S as their union. A partition, together with atotal order on the sets of the partition, gives a structure called by Richard P. Stanley an ordered partition[7] and byTheodore Motzkin a list of sets.[8] An ordered partition of a finite set may be written as a finite sequence of the setsin the partition: for instance, the three ordered partitions of the set {a, b} are

{a}, {b},{b}, {a}, and{a, b}.

In a strict weak ordering, the equivalence classes of incomparability give a set partition, in which the sets inherit atotal ordering from their elements, giving rise to an ordered partition. In the other direction, any ordered partitiongives rise to a strict weak ordering in which two elements are incomparable when they belong to the same set in thepartition, and otherwise inherit the order of the sets that contain them.

58 CHAPTER 12. WEAK ORDERING

12.2.4 Representation by functions

For sets of sufficiently small cardinality, a third axiomatization is possible, based on real-valued functions. If X is anyset and f a real-valued function on X then f induces a strict weak order on X by setting a < b if and only if f(a) < f(b).The associated total preorder is given by setting a ≲ b if and only if f(a) ≤ f(b), and the associated equivalence bysetting a ∼ b if and only if f(a) = f(b).The relations do not change when f is replaced by g o f (composite function), where g is a strictly increasing real-valued function defined on at least the range of f. Thus e.g. a utility function defines a preference relation. In thiscontext, weak orderings are also known as preferential arrangements.[9]

If X is finite or countable, every weak order on X can be represented by a function in this way.[10] However, thereexist strict weak orders that have no corresponding real function. For example, there is no such function for thelexicographic order on Rn. Thus, while in most preference relation models the relation defines a utility function up toorder-preserving transformations, there is no such function for lexicographic preferences.More generally, if X is a set, and Y is a set with a strict weak ordering "<", and f a function from X to Y, then finduces a strict weak ordering on X by setting a < b if and only if f(a) < f(b). As before, the associated total preorderis given by setting a ≲ b if and only if f(a) ≲ f(b), and the associated equivalence by setting a ∼ b if and onlyif f(a) ∼ f(b). It is not assumed here that f is an injective function, so a class of two equivalent elements on Ymay induce a larger class of equivalent elements on X. Also, f is not assumed to be an surjective function, so a classof equivalent elements on Y may induce a smaller or empty class on X. However, the function f induces an injectivefunction that maps the partition on X to that on Y. Thus, in the case of finite partitions, the number of classes in X isless than or equal to the number of classes on Y.

12.3 Related types of ordering

Semiorders generalize strict weak orderings, but do not assume transitivity of incomparability.[11] A strict weak orderthat is trichotomous is called a strict total order.[12] The total preorder which is the inverse of its complement is inthis case a total order.For a strict weak order "<" another associated reflexive relation is its reflexive closure, a (non-strict) partial order "≤".The two associated reflexive relations differ with regard to different a and b for which neither a < b nor b < a: in thetotal preorder corresponding to a strict weak order we get a ≲ b and b ≲ a, while in the partial order given by thereflexive closure we get neither a ≤ b nor b ≤ a. For strict total orders these two associated reflexive relations arethe same: the corresponding (non-strict) total order.[12] The reflexive closure of a strict weak ordering is a type ofseries-parallel partial order.

12.4 All weak orders on a finite set

12.4.1 Combinatorial enumeration

Main article: ordered Bell number

The number of distinct weak orders (represented either as strict weak orders or as total preorders) on an n-elementset is given by the following sequence (sequence A000670 in OEIS):These numbers are also called the Fubini numbers or ordered Bell numbers.For example, for a set of three labeled items, there is one weak order in which all three items are tied. There are threeways of partitioning the items into one singleton set and one group of two tied items, and each of these partitionsgives two weak orders (one in which the singleton is smaller than the group of two, and one in which this ordering isreversed), giving six weak orders of this type. And there is a single way of partitioning the set into three singletons,which can be totally ordered in six different ways. Thus, altogether, there are 13 different weak orders on three items.

12.4. ALL WEAK ORDERS ON A FINITE SET 59

(4,1,2,3)(4,2,1,3)

(3,2,1,4)

(3,1,2,4)

(2,1,3,4)

(1,2,3,4)

(1,2,4,3)

(1,3,2,4)

(2,1,4,3)

(2,3,1,4)

(3,1,4,2)

(4,1,3,2)

(4,2,3,1)

(3,2,4,1)(2,4,1,3)

(1,4,2,3)

(1,3,4,2)

(2,3,4,1)

(1,4,3,2)

(2,4,3,1)

(3,4,2,1)

(4,3,2,1)

(4,3,1,2)

(3,4,1,2)

The permutohedron on four elements, a three-dimensional convex polyhedron. It has 24 vertices, 36 edges, and 14 two-dimensionalfaces, which all together with the whole three-dimensional polyhedron correspond to the 75 weak orderings on four elements.

12.4.2 Adjacency structure

Unlike for partial orders, the family of weak orderings on a given finite set is not in general connected by moves thatadd or remove a single order relation to a given ordering. For instance, for three elements, the ordering in which allthree elements are tied differs by at least two pairs from any other weak ordering on the same set, in either the strictweak ordering or total preorder axiomatizations. However, a different kind of move is possible, in which the weakorderings on a set are more highly connected. Define a dichotomy to be a weak ordering with two equivalence classes,and define a dichotomy to be compatible with a given weak ordering if every two elements that are related in theordering are either related in the same way or tied in the dichotomy. Alternatively, a dichotomy may be defined as aDedekind cut for a weak ordering. Then a weak ordering may be characterized by its set of compatible dichotomies.For a finite set of labeled items, every pair of weak orderings may be connected to each other by a sequence of movesthat add or remove one dichotomy at a time to or from this set of dichotomies. Moreover, the undirected graph thathas the weak orderings as its vertices, and these moves as its edges, forms a partial cube.[13]

Geometrically, the total orderings of a given finite set may be represented as the vertices of a permutohedron, and thedichotomies on this same set as the facets of the permutohedron. In this geometric representation, the weak orderingson the set correspond to the faces of all different dimensions of the permutohedron (including the permutohedronitself, but not the empty set, as a face). The codimension of a face gives the number of equivalence classes in thecorresponding weak ordering.[14] In this geometric representation the partial cube of moves on weak orderings is thegraph describing the covering relation of the face lattice of the permutohedron.For instance, for n = 3, the permutohedron on three elements is just a regular hexagon. The face lattice of the hexagon(again, including the hexagon itself as a face, but not including the empty set) has thirteen elements: one hexagon,six edges, and six vertices, corresponding to the one completely tied weak ordering, six weak orderings with one tie,and six total orderings. The graph of moves on these 13 weak orderings is shown in the figure.

60 CHAPTER 12. WEAK ORDERING

12.5 Applications

As mentioned above, weak orders have applications in utility theory.[10] In linear programming and other types ofcombinatorial optimization problem, the prioritization of solutions or of bases is often given by a weak order, de-termined by a real-valued objective function; the phenomenon of ties in these orderings is called “degeneracy”, andseveral types of tie-breaking rule have been used to refine this weak ordering into a total ordering in order to preventproblems caused by degeneracy.[15]

Weak orders have also been used in computer science, in partition refinement based algorithms for lexicographicbreadth-first search and lexicographic topological ordering. In these algorithms, a weak ordering on the vertices of agraph (represented as a family of sets that partition the vertices, together with a doubly linked list providing a totalorder on the sets) is gradually refined over the course of the algorithm, eventually producing a total ordering that isthe output of the algorithm.[16]

In the Standard Library for the C++ programming language, the set and multiset data types sort their input by acomparison function that is specified at the time of template instantiation, and that is assumed to implement a strictweak ordering.[17]

12.6 References[1] Roberts, Fred; Tesman, Barry (2011), Applied Combinatorics (2nd ed.), CRC Press, Section 4.2.4 Weak Orders, pp. 254–

256, ISBN 9781420099836.

[2] de Koninck, J. M. (2009), Those Fascinating Numbers, American Mathematical Society, p. 4, ISBN 9780821886311.

[3] Baker, Kent (April 29, 2007), “The Bruce hangs on for Hunt Cup victory: Bug River, Lear Charm finish in dead heat forsecond”, The Baltimore Sun, (subscription required (help)).

[4] Regenwetter, Michel (2006), Behavioral Social Choice: Probabilistic Models, Statistical Inference, and Applications, Cam-bridge University Press, pp. 113ff, ISBN 9780521536660.

[5] Flaška, V.; Ježek, J.; Kepka, T.; Kortelainen, J. (2007). Transitive Closures of Binary Relations I (PDF). Prague: Schoolof Mathematics - Physics Charles University. p. 1. Lemma 1.1 (iv). Note that this source refers to asymmetric relationsas “strictly antisymmetric”.

[6] Ehrgott, Matthias (2005), Multicriteria Optimization, Springer, Proposition 1.9, p. 10, ISBN 9783540276593.

[7] Stanley, Richard P. (1997), Enumerative Combinatorics, Vol. 2, Cambridge Studies in Advanced Mathematics 62, Cam-bridge University Press, p. 297.

[8] Motzkin, Theodore S. (1971), “Sorting numbers for cylinders and other classification numbers”, Combinatorics (Proc.Sympos. Pure Math., Vol. XIX, Univ. California, Los Angeles, Calif., 1968), Providence, R.I.: Amer. Math. Soc., pp.167–176, MR 0332508.

[9] Gross, O. A. (1962), “Preferential arrangements”, The American Mathematical Monthly 69: 4–8, doi:10.2307/2312725,MR 0130837.

[10] Roberts, Fred S. (1979), Measurement Theory, with Applications to Decisionmaking, Utility, and the Social Sciences, Ency-clopedia of Mathematics and its Applications 7, Addison-Wesley, Theorem 3.1, ISBN 978-0-201-13506-0.

[11] Luce, R. Duncan (1956), “Semiorders and a theory of utility discrimination”, Econometrica 24: 178–191, JSTOR 1905751,MR 0078632.

[12] Velleman, Daniel J. (2006),How to Prove It: A Structured Approach, Cambridge University Press, p. 204, ISBN 9780521675994.

[13] Eppstein, David; Falmagne, Jean-Claude; Ovchinnikov, Sergei (2008), Media Theory: Interdisciplinary Applied Mathemat-ics, Springer, Section 9.4, Weak Orders and Cubical Complexes, pp. 188–196.

[14] Ziegler, Günter M. (1995), Lectures on Polytopes, Graduate Texts in Mathematics 152, Springer, p. 18.

[15] Chvátal, Vašek (1983), Linear Programming, Macmillan, pp. 29–38, ISBN 9780716715870.

[16] Habib, Michel; Paul, Christophe; Viennot, Laurent (1999), “Partition refinement techniques: an interesting algorithmictool kit”, International Journal of Foundations of Computer Science 10 (2): 147–170, doi:10.1142/S0129054199000125,MR 1759929.

[17] Josuttis, Nicolai M. (2012), The C++ Standard Library: A Tutorial and Reference, Addison-Wesley, p. 469, ISBN9780132977739.

Chapter 13

Well-founded relation

“Noetherian induction” redirects here. For the use in topology, see Noetherian topological space.

In mathematics, a binary relation, R, is well-founded (or wellfounded) on a class X if and only if every non-emptysubset S X has a minimal element; that is, some element m of any S is not related by sRm (for instance, "m is notsmaller than”) for the rest of the s ∈ S.

∀S ⊆ X (S ̸= ∅ → ∃m ∈ S ∀s ∈ S (s,m) /∈ R)

(Some authors include an extra condition that R is set-like, i.e., that the elements less than any given element form aset.)Equivalently, assuming some choice, a relation is well-founded if and only if it contains no countable infinite descend-ing chains: that is, there is no infinite sequence x0, x1, x2, ... of elements of X such that xn₊₁ R x for every naturalnumber n.In order theory, a partial order is called well-founded if the corresponding strict order is a well-founded relation. Ifthe order is a total order then it is called a well-order.In set theory, a set x is called a well-founded set if the set membership relation is well-founded on the transitiveclosure of x. The axiom of regularity, which is one of the axioms of Zermelo–Fraenkel set theory, asserts that all setsare well-founded.A relation R is converse well-founded, upwards well-founded or Noetherian on X, if the converse relation R−1

is well-founded on X. In this case R is also said to satisfy the ascending chain condition. In the context of rewritingsystems, a Noetherian relation is also called terminating.

13.1 Induction and recursion

An important reason that well-founded relations are interesting is because a version of transfinite induction can beused on them: if (X, R) is a well-founded relation, P(x) is some property of elements of X, and we want to show that

P(x) holds for all elements x of X,

it suffices to show that:

If x is an element of X and P(y) is true for all y such that y R x, then P(x) must also be true.

That is,

∀x ∈ X [(∀y ∈ X (y Rx → P (y))) → P (x)] → ∀x ∈ X P (x).

61

62 CHAPTER 13. WELL-FOUNDED RELATION

Well-founded induction is sometimes called Noetherian induction,[1] after Emmy Noether.On par with induction, well-founded relations also support construction of objects by transfinite recursion. Let (X,R) be a set-like well-founded relation and F a function that assigns an object F(x, g) to each pair of an element x ∈ Xand a function g on the initial segment {y: y R x} of X. Then there is a unique function G such that for every x ∈ X,

G(x) = F (x,G|{y:y Rx})

That is, if we want to construct a function G on X, we may define G(x) using the values of G(y) for y R x.As an example, consider the well-founded relation (N, S), where N is the set of all natural numbers, and S is the graphof the successor function x → x + 1. Then induction on S is the usual mathematical induction, and recursion on Sgives primitive recursion. If we consider the order relation (N, <), we obtain complete induction, and course-of-valuesrecursion. The statement that (N, <) is well-founded is also known as the well-ordering principle.There are other interesting special cases of well-founded induction. When the well-founded relation is the usualordering on the class of all ordinal numbers, the technique is called transfinite induction. When the well-founded setis a set of recursively-defined data structures, the technique is called structural induction. When the well-foundedrelation is set membership on the universal class, the technique is known as ∈-induction. See those articles for moredetails.

13.2 Examples

Well-founded relations which are not totally ordered include:

• the positive integers {1, 2, 3, ...}, with the order defined by a < b if and only if a divides b and a ≠ b.

• the set of all finite strings over a fixed alphabet, with the order defined by s < t if and only if s is a propersubstring of t.

• the set N × N of pairs of natural numbers, ordered by (n1, n2) < (m1, m2) if and only if n1 < m1 and n2 < m2.

• the set of all regular expressions over a fixed alphabet, with the order defined by s < t if and only if s is a propersubexpression of t.

• any class whose elements are sets, with the relation ∈ (“is an element of”). This is the axiom of regularity.

• the nodes of any finite directed acyclic graph, with the relation R defined such that a R b if and only if there isan edge from a to b.

Examples of relations that are not well-founded include:

• the negative integers {−1, −2, −3, …}, with the usual order, since any unbounded subset has no least element.

• The set of strings over a finite alphabet with more than one element, under the usual (lexicographic) order,since the sequence “B” > “AB” > “AAB” > “AAAB” > … is an infinite descending chain. This relation failsto be well-founded even though the entire set has a minimum element, namely the empty string.

• the rational numbers (or reals) under the standard ordering, since, for example, the set of positive rationals (orreals) lacks a minimum.

13.3 Other properties

If (X, <) is a well-founded relation and x is an element of X, then the descending chains starting at x are all finite, butthis does not mean that their lengths are necessarily bounded. Consider the following example: Let X be the union ofthe positive integers and a new element ω, which is bigger than any integer. Then X is a well-founded set, but thereare descending chains starting at ω of arbitrary great (finite) length; the chain ω, n − 1, n − 2, ..., 2, 1 has length nfor any n.

13.4. REFLEXIVITY 63

The Mostowski collapse lemma implies that set membership is a universal among the extensional well-founded rela-tions: for any set-like well-founded relation R on a class X which is extensional, there exists a class C such that (X,R)is isomorphic to (C,∈).

13.4 Reflexivity

A relation R is said to be reflexive if a R a holds for every a in the domain of the relation. Every reflexive relation on anonempty domain has infinite descending chains, because any constant sequence is a descending chain. For example,in the natural numbers with their usual order ≤, we have 1 ≥ 1 ≥ 1 ≥ · · · . To avoid these trivial descendingsequences, when working with a reflexive relation R it is common to use (perhaps implicitly) the alternate relation R′defined such that a R′ b if and only if a R b and a ≠ b. In the context of the natural numbers, this means that therelation <, which is well-founded, is used instead of the relation ≤, which is not. In some texts, the definition of awell-founded relation is changed from the definition above to include this convention.

13.5 References[1] Bourbaki, N. (1972) Elements of mathematics. Commutative algebra, Addison-Wesley.

• Just, Winfried and Weese, Martin, Discovering Modern Set theory. I, American Mathematical Society (1998)ISBN 0-8218-0266-6.

Chapter 14

Well-order

In mathematics, a well-order relation (or well-ordering) on a set S is a total order on S with the property that everynon-empty subset of S has a least element in this ordering. The set S together with the well-order relation is thencalled awell-ordered set. The hyphen is frequently omitted in contemporary papers, yielding the spellingswellorder,wellordered, and wellordering.Every non-empty well-ordered set has a least element. Every element s of a well-ordered set, except a possible greatestelement, has a unique successor (next element), namely the least element of the subset of all elements greater thans. There may be elements besides the least element which have no predecessor (see Natural numbers below for anexample). In a well-ordered set S, every subset T which has an upper bound has a least upper bound, namely the leastelement of the subset of all upper bounds of T in S.If ≤ is a non-strict well-ordering, then < is a strict well-ordering. A relation is a strict well-ordering if and only if it isa well-founded strict total order. The distinction between strict and non-strict well-orders is often ignored since theyare easily interconvertible.Every well-ordered set is uniquely order isomorphic to a unique ordinal number, called the order type of the well-ordered set. The well-ordering theorem, which is equivalent to the axiom of choice, states that every set can bewell-ordered. If a set is well-ordered (or even if it merely admits a wellfounded relation), the proof technique oftransfinite induction can be used to prove that a given statement is true for all elements of the set.The observation that the natural numbers are well-ordered by the usual less-than relation is commonly called thewell-ordering principle (for natural numbers).

14.1 Ordinal numbers

Main article: Ordinal number

Every well-ordered set is uniquely order isomorphic to a unique ordinal number, called the order type of the well-ordered set. The position of each element within the ordered set is also given by an ordinal number. In the case of afinite set, the basic operation of counting, to find the ordinal number of a particular object, or to find the object witha particular ordinal number, corresponds to assigning ordinal numbers one by one to the objects. The size (numberof elements, cardinal number) of a finite set is equal to the order type. Counting in the everyday sense typically startsfrom one, so it assigns to each object the size of the initial segment with that object as last element. Note that thesenumbers are one more than the formal ordinal numbers according to the isomorphic order, because these are equalto the number of earlier objects (which corresponds to counting from zero). Thus for finite n, the expression "n-thelement” of a well-ordered set requires context to know whether this counts from zero or one. In a notation "β-thelement” where β can also be an infinite ordinal, it will typically count from zero.For an infinite set the order type determines the cardinality, but not conversely: well-ordered sets of a particularcardinality can have many different order types. For a countably infinite set, the set of possible order types is evenuncountable.

64

14.2. EXAMPLES AND COUNTEREXAMPLES 65

14.2 Examples and counterexamples

14.2.1 Natural numbers

The standard ordering ≤ of the natural numbers is a well-ordering and has the additional property that every non-zeronatural number has a unique predecessor.Another well-ordering of the natural numbers is given by defining that all even numbers are less than all odd numbers,and the usual ordering applies within the evens and the odds:

0 2 4 6 8 ... 1 3 5 7 9 ...

This is a well-ordered set of order type ω + ω. Every element has a successor (there is no largest element). Twoelements lack a predecessor: 0 and 1.

14.2.2 Integers

Unlike the standard ordering ≤ of the natural numbers, the standard ordering ≤ of the integers is not a well-ordering,since, for example, the set of negative integers does not contain a least element.The following relation R is an example of well-ordering of the integers: x R y if and only if one of the followingconditions holds:

1. x = 0

2. x is positive, and y is negative

3. x and y are both positive, and x ≤ y

4. x and y are both negative, and |x| ≤ |y|

This relation R can be visualized as follows:

0 1 2 3 4 ... −1 −2 −3 ...

R is isomorphic to the ordinal number ω + ω.Another relation for well-ordering the integers is the following definition: x ≤ y iff (|x| < |y| or (|x| = |y| and x ≤ y)).This well-order can be visualized as follows:

0 −1 1 −2 2 −3 3 −4 4 ...

This has the order type ω.

14.2.3 Reals

The standard ordering ≤ of any real interval is not a well-ordering, since, for example, the open interval (0, 1) ⊆ [0,1]does not contain a least element. From the ZFC axioms of set theory (including the axiom of choice) one can showthat there is a well-order of the reals. Also Wacław Sierpiński proved that ZF + GCH (the generalized continuumhypothesis) imply the axiom of choice and hence a well-order of the reals. Nonetheless, it is possible to show thatthe ZFC+GCH axioms alone are not sufficient to prove the existence of a definable (by a formula) well-order ofthe reals.[1] However it is consistent with ZFC that a definable well-ordering of the reals exists—for example, it isconsistent with ZFC that V=L, and it follows from ZFC+V=L that a particular formula well-orders the reals, or indeedany set.An uncountable subset of the real numbers with the standard ordering ≤ cannot be a well-order: SupposeX is a subsetof R well-ordered by ≤. For each x in X, let s(x) be the successor of x in ≤ ordering on X (unless x is the last elementof X). Let A = { (x, s(x)) | x ∈ X } whose elements are nonempty and disjoint intervals. Each such interval contains

66 CHAPTER 14. WELL-ORDER

at least one rational number, so there is an injective function from A to Q. There is an injection from X to A (exceptpossibly for a last element of X which could be mapped to zero later). And it is well known that there is an injectionfrom Q to the natural numbers (which could be chosen to avoid hitting zero). Thus there is an injection from X to thenatural numbers which means that X is countable. On the other hand, a countably infinite subset of the reals may ormay not be a well-order with the standard "≤".

• The natural numbers are a well-order.

• The set {1/n : n =1,2,3,...} has no least element and is therefore not a well-order (again, under standard ordering≤).

Examples of well-orders:

• The set of numbers { − 2−n | 0 ≤ n < ω } has order type ω.

• The set of numbers { − 2−n − 2−m−n | 0 ≤ m,n < ω } has order type ω². The previous set is the set of limitpoints within the set. Within the set of real numbers, either with the ordinary topology or the order topology,0 is also a limit point of the set. It is also a limit point of the set of limit points.

• The set of numbers { − 2−n | 0 ≤ n < ω } ∪ { 1 } has order type ω + 1. With the order topology of this set, 1 isa limit point of the set. With the ordinary topology (or equivalently, the order topology) of the real numbers itis not.

14.3 Equivalent formulations

If a set is totally ordered, then the following are equivalent to each other:

1. The set is well-ordered. That is, every nonempty subset has a least element.

2. Transfinite induction works for the entire ordered set.

3. Every strictly decreasing sequence of elements of the set must terminate after only finitely many steps (assumingthe axiom of dependent choice).

4. Every subordering is isomorphic to an initial segment.

14.4 Order topology

Every well-ordered set can be made into a topological space by endowing it with the order topology.With respect to this topology there can be two kinds of elements:

• isolated points - these are the minimum and the elements with a predecessor.

• limit points - this type does not occur in finite sets, and may or may not occur in an infinite set; the infinite setswithout limit point are the sets of order type ω, for example N.

For subsets we can distinguish:

• Subsets with a maximum (that is, subsets which are bounded by themselves); this can be an isolated point or alimit point of the whole set; in the latter case it may or may not be also a limit point of the subset.

• Subsets which are unbounded by themselves but bounded in the whole set; they have no maximum, but asupremum outside the subset; if the subset is non-empty this supremum is a limit point of the subset and hencealso of the whole set; if the subset is empty this supremum is the minimum of the whole set.

• Subsets which are unbounded in the whole set.

14.5. SEE ALSO 67

A subset is cofinal in the whole set if and only if it is unbounded in the whole set or it has a maximum which is alsomaximum of the whole set.A well-ordered set as topological space is a first-countable space if and only if it has order type less than or equal toω1 (omega-one), that is, if and only if the set is countable or has the smallest uncountable order type.

14.5 See also• Tree (set theory), generalization

• Well-ordering theorem

• Ordinal number

• Well-founded set

• Well partial order

• Prewellordering

• Directed set

14.6 References[1] S. Feferman: “Some Applications of the Notions of Forcing and Generic Sets”, Fundamenta Mathematicae, 56 (1964)

325-345

• Folland, Gerald B. (1999). Real Analysis: Modern Techniques and Their Applications. Pure and applied math-ematics (2nd ed.). John Wiley & Sons. pp. 4–6, 9. ISBN 978-0-471-31716-6.

Chapter 15

Well-quasi-ordering

In mathematics, specifically order theory, a well-quasi-ordering or wqo is a quasi-ordering such that any infinitesequence of elements x0 , x1 , x2 , … from X contains an increasing pair xi ≤ xj with i < j .

15.1 Motivation

Well-founded induction can be used on any set with a well-founded relation, thus one is interested in when a quasi-order is well-founded. However the class of well-founded quasiorders is not closed under certain operations - thatis, when a quasi-order is used to obtain a new quasi-order on a set of structures derived from our original set, thisquasiorder is found to be not well-founded. By placing stronger restrictions on the original well-founded quasiorderingone can hope to ensure that our derived quasiorderings are still well-founded.An example of this is the power set operation. Given a quasiordering ≤ for a set X one can define a quasiorder ≤+

on X 's power set P (X) by setting A ≤+ B if and only if for each element of A one can find some element of Bwhich is larger than it under ≤ . One can show that this quasiordering on P (X) needn't be well-founded, but if onetakes the original quasi-ordering to be a well-quasi-ordering, then it is.

15.2 Formal definition

A well-quasi-ordering on a set X is a quasi-ordering (i.e., a reflexive, transitive binary relation) such that any infinitesequence of elements x0 , x1 , x2 , … from X contains an increasing pair xi ≤ xj with i < j . The set X is said tobe well-quasi-ordered, or shortly wqo.A well partial order, or a wpo, is a wqo that is a proper ordering relation, i.e., it is antisymmetric.Among other ways of defining wqo’s, one is to say that they are quasi-orderings which do not contain infinite strictlydecreasing sequences (of the form x0 > x1 > x2 >…) nor infinite sequences of pairwise incomparable elements.Hence a quasi-order ( X ,≤) is wqo if and only if it is well-founded and has no infinite antichains.

15.3 Examples

• (N,≤) , the set of natural numbers with standard ordering, is a well partial order (in fact, a well-order). How-ever, (Z,≤) , the set of positive and negative integers, is not a well-quasi-order, because it is not well-founded.

• (N, |) , the set of natural numbers ordered by divisibility, is not a well partial order: the prime numbers are aninfinite antichain.

• (Nk,≤) , the set of vectors of k natural numbers (where k is finite) with component-wise ordering, is a wellpartial order (Dickson’s lemma). More generally, if (X,≤) is well-quasi-order, then (Xk,≤k) is also a well-quasi-order for all k .

68

15.4. WQO’S VERSUS WELL PARTIAL ORDERS 69

• LetX be an arbitrary finite set with at least two elements. The setX∗ of words overX ordered lexicographically(as in a dictionary) isnot a well-quasi-order because it contains the infinite decreasing sequence b, ab, aab, aaab, . . .. Similarly, X∗ ordered by the prefix relation is not a well-quasi-order, because the previous sequence is an in-finite antichain of this partial order. However, X∗ ordered by the subsequence relation is a well partial order.[1]

(If X has only one element, these three partial orders are identical.)

• More generally, (X∗,≤) , the set of finite X -sequences ordered by embedding is a well-quasi-order if andonly if (X,≤) is a well-quasi-order (Higman’s lemma). Recall that one embeds a sequence u into a sequence vby finding a subsequence of v that has the same length as u and that dominates it term by term. When (X,=)is a finite unordered set, u ≤ v if and only if u is a subsequence of v .

• (Xω,≤) , the set of infinite sequences over a well-quasi-order (X,≤) , ordered by embedding, is not a well-quasi-order in general. That is, Higman’s lemma does not carry over to infinite sequences. Better-quasi-orderings have been introduced to generalize Higman’s lemma to sequences of arbitrary lengths.

• Embedding between finite trees with nodes labeled by elements of a wqo (X,≤) is a wqo (Kruskal’s treetheorem).

• Embedding between infinite trees with nodes labeled by elements of a wqo (X,≤) is a wqo (Nash-Williams'theorem).

• Embedding between countable scattered linear order types is a well-quasi-order (Laver's theorem).

• Embedding between countable boolean algebras is a well-quasi-order. This follows from Laver’s theorem anda theorem of Ketonen.

• Finite graphs ordered by a notion of embedding called "graph minor" is a well-quasi-order (Robertson–Seymourtheorem).

• Graphs of finite tree-depth ordered by the induced subgraph relation form a well-quasi-order,[2] as do thecographs ordered by induced subgraphs.[3]

15.4 Wqo’s versus well partial orders

In practice, the wqo’s one manipulates are quite often not orderings (see examples above), and the theory is technicallysmoother if we do not require antisymmetry, so it is built with wqo’s as the basic notion.Observe that a wpo is a wqo, and that a wqo gives rise to a wpo between equivalence classes induced by the kernelof the wqo. For example, if we order Z by divisibility, we end up with n ≡ m if and only if n = ±m , so that(Z, |) ≈ (N, |) .

15.5 Infinite increasing subsequences

If ( X , ≤) is wqo then every infinite sequence x0 , x1 , x2 , … contains an infinite increasing subsequence xn0 ≤xn1 ≤ xn2 ≤… (with n0 < n1 < n2 <…). Such a subsequence is sometimes called perfect. This can be proved bya Ramsey argument: given some sequence (xi)i , consider the set I of indexes i such that xi has no larger or equalxj to its right, i.e., with i < j . If I is infinite, then the I -extracted subsequence contradicts the assumption that Xis wqo. So I is finite, and any xn with n larger than any index in I can be used as the starting point of an infiniteincreasing subsequence.The existence of such infinite increasing subsequences is sometimes taken as a definition for well-quasi-ordering,leading to an equivalent notion.

15.6 Properties of wqos

• Given a quasiordering (X,≤) the quasiordering (P (X),≤+) defined by A ≤+ B ⇐⇒ ∀a ∈ A∃b ∈ B(a ≤b) is well-founded if and only if (X,≤) is a wqo.[4]

70 CHAPTER 15. WELL-QUASI-ORDERING

• A quasiordering is a wqo if and only if the corresponding partial order (obtained by quotienting by x ∼ y ⇐⇒x ≤ y ∧ y ≤ x ) has no infinite descending sequences or antichains. (This can be proved using a Ramseyargument as above.)

• Given a well-quasi-ordering (X,≤) , any sequence of subsets S0 ⊆ S1 ⊆ ... ⊆ X such that ∀i ∈ N, ∀x, y ∈X,x ≤ y∧x ∈ Si ⇒ y ∈ Si eventually stabilises (meaning there is an indexn ∈ N such thatSn = Sn+1 = ...; subsets S ⊆ X with the property ∀x, y ∈ X,x ≤ y ∧ x ∈ S ⇒ y ∈ S are usually called upward-closed):assuming the contrary ∀i ∈ N∃j ∈ N, j > i, ∃x ∈ Sj \Si , a contradiction is reached by extracting an infinitenon-ascending subsequence.

• Given a well-quasi-ordering (X,≤) , any subset S ⊆ X which is upward-closed with respect to ≤ has a finitenumber of minimal elements w.r.t. ≤ , for otherwise the minimal elements of S would constitute an infiniteantichain.

15.7 Notes[1] Gasarch, W. (1998), “A survey of recursive combinatorics”, Handbook of Recursive Mathematics, Vol. 2, Stud. Logic

Found. Math. 139, Amsterdam: North-Holland, pp. 1041–1176, doi:10.1016/S0049-237X(98)80049-9, MR 1673598.See in particular page 1160.

[2] Nešetřil, Jaroslav; Ossona de Mendez, Patrice (2012), “Lemma 6.13”, Sparsity: Graphs, Structures, and Algorithms, Algo-rithms and Combinatorics 28, Heidelberg: Springer, p. 137, doi:10.1007/978-3-642-27875-4, ISBN 978-3-642-27874-7,MR 2920058.

[3] Damaschke, Peter (1990), “Induced subgraphs and well-quasi-ordering”, Journal of Graph Theory 14 (4): 427–435,doi:10.1002/jgt.3190140406, MR 1067237.

[4] Forster, Thomas (2003). “Better-quasi-orderings and coinduction”. Theoretical Computer Science 309 (1–3): 111–123.doi:10.1016/S0304-3975(03)00131-2.

15.8 References• Dickson, L. E. (1913). “Finiteness of the odd perfect and primitive abundant numbers with r distinct prime

factors”. American Journal of Mathematics 35 (4): 413–422. doi:10.2307/2370405. JSTOR 2370405.

• Higman, G. (1952). “Ordering by divisibility in abstract algebras”. Proceedings of the London MathematicalSociety 2: 326–336. doi:10.1112/plms/s3-2.1.326.

• Kruskal, J. B. (1972). “The theory of well-quasi-ordering: A frequently discovered concept”. Journal ofCombinatorial Theory. Series A 13 (3): 297–305. doi:10.1016/0097-3165(72)90063-5.

• Ketonen, Jussi (1978). “The structure of countable Boolean algebras”. Annals of Mathematics 108 (1): 41–89.doi:10.2307/1970929. JSTOR 1970929.

• Milner, E. C. (1985). “Basic WQO- and BQO-theory”. In Rival, I. Graphs and Order. The Role of Graphs inthe Theory of Ordered Sets and Its Applications. D. Reidel Publishing Co. pp. 487–502. ISBN 90-277-1943-8.

• Gallier, Jean H. (1991). “What’s so special about Kruskal’s theorem and the ordinal Γo? A survey of some re-sults in proof theory”. Annals of Pure and Applied Logic 53 (3): 199–260. doi:10.1016/0168-0072(91)90022-E.

15.9 See also• Better-quasi-ordering

• Prewellordering

• Well-order

15.10. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 71

15.10 Text and image sources, contributors, and licenses

15.10.1 Text• Binary relation Source: https://en.wikipedia.org/wiki/Binary_relation?oldid=695052935 Contributors: AxelBoldt, Bryan Derksen, Zun-

dark, Tarquin, Jan Hidders, Roadrunner, Mjb, Tomo, Patrick, Xavic69, Michael Hardy, Wshun, Isomorphic, Dominus, Ixfd64, Takuya-Murata, Charles Matthews, Timwi, Dcoetzee, Jitse Niesen, Robbot, Chocolateboy, MathMartin, Tobias Bergemann, Giftlite, Fropuff,Dratman, Jorge Stolfi, Jlr~enwiki, Andycjp, Quarl, Guanabot, Yuval madar, Slipstream, Paul August, Elwikipedista~enwiki, Shanes,EmilJ, Randall Holmes, Ardric47, Obradovic Goran, Eje211, Alansohn, Dallashan~enwiki, Keenan Pepper, PAR, Adrian.benko, OlegAlexandrov, Joriki, Linas, MFH, Dpv, Pigcatian, Penumbra2000, Fresheneesz, Chobot, YurikBot, Hairy Dude, Koffieyahoo, Trova-tore, Bota47, Arthur Rubin, Netrapt, SmackBot, Royalguard11, SEIBasaurus, Cybercobra, Jon Awbrey, Turms, Lambiam, Dbtfz, MrStephen, Mets501, Dreftymac, Happy-melon, Petr Matas, CRGreathouse, CBM, Yrodro, WillowW, Xantharius, Thijs!bot, Egriffin,Rlupsa, JAnDbot, MER-C, Magioladitis, Vanish2, Avicennasis, David Eppstein, Robin S, Akurn, Adavidb, LajujKej, Owlgorithm, Djjrjr,Policron, DavidCBryant, Quux0r, VolkovBot, Boute, Vipinhari, Anonymous Dissident, PaulTanenbaum, Jackfork, Wykypydya, Dmcq,AlleborgoBot, AHMartin, Ocsenave, Sftd, Paradoctor, Henry Delforn (old), MiNombreDeGuerra, DuaneLAnderson, Anchor Link Bot,CBM2, Classicalecon, ClueBot, Snigbrook, Rhubbarb, Hans Adler, SilvonenBot, BYS2, Plmday, Addbot, LinkFA-Bot, Tide rolls, Jar-ble, Legobot, Luckas-bot, Yobot, Ht686rg90, Pcap, Labus, Nallimbot, Reindra, FredrikMeyer, AnomieBOT, Floquenbeam, Royote,Hahahaha4, Materialscientist, Belkovich, Citation bot, Racconish, Jellystones, Xqbot, Isheden, Geero, GhalyBot, Ernsts, Howard Mc-Cay, Constructive editor, Mark Renier, Mfwitten, RandomDSdevel, NearSetAccount, SpaceFlight89, Yunshui, Miracle Pen, Bramble-clawx, RjwilmsiBot, Nomen4Omen, Chharvey, SporkBot, OnePt618, Sameer143, Socialservice, ResearchRave, ClueBot NG, Wcherowi,Frietjes, Helpful Pixie Bot, Koertefa, ChrisGualtieri, YFdyh-bot, Dexbot, Makecat-bot, ScitDei, Lerutit, Jochen Burghardt, Jodosma,Karim132, Monkbot, Pratincola, , Some1Redirects4You, The Quixotic Potato and Anonymous: 103

• Intransitivity Source: https://en.wikipedia.org/wiki/Intransitivity?oldid=678086296 Contributors: Michael Hardy, Booyabazooka, Rp,6birc, Radicalsubversiv, Andres, Charles Matthews, Ruakh, Giftlite, Andris, Quickwik, D6, Xezbeth, Paul August, Jnestorius, Spoon!,Polluks, SmackBot, Melchoir, Mauls, MisterHand, Lambiam, Iamagloworm, Dinkumator, CRGreathouse, CmdrObot, CBM, Thomas-meeks, Ael 2, Thijs!bot, VoABot II, Cnilep, Ddxc, Anchor Link Bot, PixelBot, DumZiBoT, Pa68, Addbot, Forich, WissensDürster,Undsoweiter, HRoestBot, ChronoKinetic, RjwilmsiBot, Chharvey, Diakov and Anonymous: 9

• Near sets Source: https://en.wikipedia.org/wiki/Near_sets?oldid=680626580 Contributors: Michael Hardy, Bearcat, Gandalf61, Giftlite,Rjwilmsi, Algebraist, SmackBot, MartinPoulter, Headbomb, NSH001, Sebras, Salih, Philip Trueman, JL-Bot, SchreiberBike, Tassede-the, Jarble, Alvin Seville, Erik9bot, FrescoBot, NSH002, NearSetAccount, AManWithNoPlan, Frietjes, Helpful Pixie Bot, BG19bot,ChristopherJamesHenry and Anonymous: 2

• Preorder Source: https://en.wikipedia.org/wiki/Preorder?oldid=680540484 Contributors: AxelBoldt, Patrick, Repton, Delirium, An-dres, Dysprosia, Greenrd, Big Bob the Finder, BenRG, Tobias Bergemann, Giftlite, Markus Krötzsch, Lethe, Fropuff, Vadmium, De-fLog~enwiki, Zzo38, Jh51681, Barnaby dawson, Paul August, EmilJ, Msh210, Melaen, Joriki, Linas, Dionyziz, Mandarax, Salix alba,Cjoev, VKokielov, Mathbot, Jrtayloriv, YurikBot, Laurentius, Hairy Dude, WikidSmaht, Trovatore, Modify, Netrapt, Wasseralm, Smack-Bot, XudongGuan~enwiki, DCary, Jdthood, Mets501, PaulGS, Stotr~enwiki, Zero sharp, CRGreathouse, Michael A. White, Magioladitis,David Eppstein, Jwuthe2, Largoplazo, PaulTanenbaum, SieBot, Thehotelambush, MenoBot, Functor salad, He7d3r, Sun Creator, Cenar-ium, 1ForTheMoney, Palnot, Мыша, Legobot, Luckas-bot, AnomieBOT, DannyAsher, Xqbot, VladimirReshetnikov, BercherP, Com-putScientist, BrideOfKripkenstein, Notedgrant, WikitanvirBot, Lclem, Dfabera, SporkBot, Paolo Lipparini, RichardMills65, Khazar2,Lerutit, Jochen Burghardt, Reatlas, Damonamc and Anonymous: 26

• Prewellordering Source: https://en.wikipedia.org/wiki/Prewellordering?oldid=488790287Contributors: Zundark, Patrick, Charles Matthews,EmilJ, Oleg Alexandrov, NickBush24, Trovatore, Tetracube, That Guy, From That Show!, SmackBot, Nbarth, Henry Delforn (old), Pal-not, Citation bot and Anonymous: 3

• Quasitransitive relation Source: https://en.wikipedia.org/wiki/Quasitransitive_relation?oldid=625289154Contributors: Patrick, CharlesMatthews, Giftlite, Btyner, Salix alba, SmackBot, Zahid Abdassabur, CRGreathouse, CBM, Gregbard, Sam Staton, Erik9bot, Deltahe-dron, Jochen Burghardt and Anonymous: 1

• Quotient by an equivalence relation Source: https://en.wikipedia.org/wiki/Quotient_by_an_equivalence_relation?oldid=657098718Contributors: Michael Hardy, TakuyaMurata, Bearcat, D.Lazard, K9re11, Iaritmioawp and Fimatic

• Reduct Source: https://en.wikipedia.org/wiki/Reduct?oldid=618205741 Contributors: John Baez, Spring Rubber, SmackBot, VaughanPratt, Tikiwont, Hans Adler, Backslash Forwardslash, Gf uip, EmausBot, Bgeron, Monkbot and Anonymous: 3

• Semiorder Source: https://en.wikipedia.org/wiki/Semiorder?oldid=607501955 Contributors: Rjwilmsi, Headbomb, David Eppstein,Undsoweiter, Joel B. Lewis, Lerutit and Anonymous: 2

• Separoid Source: https://en.wikipedia.org/wiki/Separoid?oldid=588579933 Contributors: Michael Hardy, Salix alba, SmackBot, Mhym,Gregbard, Magioladitis, Qworty, Hans Adler and Strausz~enwiki

• Series-parallel partial order Source: https://en.wikipedia.org/wiki/Series-parallel_partial_order?oldid=629384776 Contributors: Za-slav, A3nm, David Eppstein, Trappist the monk, Helpful Pixie Bot, Deltahedron and Anonymous: 1

• Weak ordering Source: https://en.wikipedia.org/wiki/Weak_ordering?oldid=640088882 Contributors: Patrick, Michael Hardy, Chinju,Dcoetzee, MathMartin, Tobias Bergemann, Pretzelpaws, AlphaEtaPi, Zaslav, Aisaac, YurikBot, Gadget850, Modify, Oli Filth, Jdthood,Chlewbot, Radiant chains, Jafet, CRGreathouse, Sdorrance, Gregbard, Widefox, Medinoc, Zeitlupe, David Eppstein, Jonathanrcoxhead,Watchduck, Addbot, Kne1p, Forich, Citation bot, ArthurBot, Howard McCay, Citation bot 1, SporkBot, Joel B. Lewis, Helpful Pixie Bot,Jochen Burghardt, JustBerry and Anonymous: 12

• Well-founded relation Source: https://en.wikipedia.org/wiki/Well-founded_relation?oldid=659840399 Contributors: The Anome, Ap,Michael Hardy, Dominus, TakuyaMurata, Cyp, Charles Matthews, VeryVerily, Aleph4, Mountain, Tobias Bergemann, Filemon, Marekpetrik,Lethe, Lupin, Waltpohl, Mani1, Paul August, EmilJ, Nahabedere, MZMcBride, R.e.b., FlaBot, Trovatore, Mikeblas, Crasshopper, Mar-lasdad, That Guy, From That Show!, SmackBot, Ron.garcia, Nbarth, Mmehdi.g, Mets501, CBM, WillowW, Pgagge, JAnDbot, Albmont,Leyo, Reedy Bot, Alexsmail, Thehotelambush, Mutilin, Addbot, KamikazeBot, RibotBOT, FrescoBot, Involutive-revolution, MerlIwBot,Wohlfundi, YiFeiBot, Some1Redirects4You and Anonymous: 22

72 CHAPTER 15. WELL-QUASI-ORDERING

• Well-order Source: https://en.wikipedia.org/wiki/Well-order?oldid=688185904 Contributors: AxelBoldt, Mav, Zundark, Josh Grosse,Patrick, Michael Hardy, David Martland, Dominus, TakuyaMurata, Andres, Vargenau, Revolver, Charles Matthews, Timwi, Populus,Aleph4, R3m0t, MathMartin, Tobias Bergemann, Tosha, Giftlite, Dbenbenn, Ian Maxwell, Lethe, Arturus~enwiki, Jorend, Karl-Henner,Rich Farmbrough, Luqui, Paul August, Sligocki, RJFJR, Eyu100, Salix alba, FlaBot, Margosbot~enwiki, Chobot, YurikBot, Hairy Dude,Trovatore, Obey, Bota47, Arthur Rubin, MullerHolk, Ghazer~enwiki, GrinBot~enwiki, KnightRider~enwiki, Alan McBeth, Gelingvistoj,Mhss, Nbarth, Loodog, Jim.belk, Loadmaster, JRSpriggs, CBM, WeggeBot, Myasuda, Thijs!bot, Nadav1, Escarbot, Albmont, Odexios,VolkovBot, Don4of4, SieBot, Rumping, Fyyer, Bender2k14, His Wikiness, Palnot, Addbot, Luckas-bot, Yobot, Ptbotgourou, Jarmiz,Xqbot, GrouchoBot, Miyagawa, Adrionwells, Mitizhi, RjwilmsiBot, Honestrosewater, Hunterbd, SporkBot, Misshamid, ChrisGualtieri,Khazar2, Jose Brox, Wicklet and Anonymous: 29

• Well-quasi-ordering Source: https://en.wikipedia.org/wiki/Well-quasi-ordering?oldid=687500638Contributors: Patrick, Chinju, CharlesMatthews, Tobias Bergemann, Peter Kwok, Rich Farmbrough, Paul August, EmilJ, R.e.b., Open2universe, PhS, That Guy, From ThatShow!, Mets501, Pierre de Lyon, David Eppstein, Kope, R'n'B, Alexwright, Fcarreiro, Niceguyedc, Palnot, Addbot, DOI bot, Citationbot, FrescoBot, Citation bot 1, Gongfarmerzed, John of Reading, ZéroBot, Ɯ, Mastergreg82, Paolo Lipparini, CitationCleanerBot, JeffErickson, Mark viking, Quenhitran, Anrnusna, ,תמשל Gasarch and Anonymous: 6

15.10.2 Images• File:13-Weak-Orders.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/3c/13-Weak-Orders.svg License: Public do-

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