independent axioms for convexity
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Journal of Geometry. Vol. 5/1 1974. Birkh~user Verlag Basel
INDEPENDENT AXIOMS FOR CONVEXITY
Victor Bryant
Join-structures or Convexity Spaces generalise the geometry of Vector Spaces by means of axioms concerning line segments. Most other generalisations of this type are just particular examples of Convexity Bpaces. In the many papers on this subject the collection of axioms is too long: in this short note we exhibit an independent set of axioms for these structures.
Let X be a non-empty set, let a,b,.., be elements
of X and A,B,... subsets of X. We do not distinguish
between an element of X and the singleton subset which
it defines. Thus in X the notation ~ is redundant and
is replaced by ~ . Also we write A = B to mean A
meets B or A ~ B + ~. A join is a mapping
�9 : X x X + 2 X, i.e. it associates with each ordered
pair of elements of X a subset a.b (or simply ab) of X.
Given a join we can define a new operation
/ : X x X § 2 X by a/b = {x : a Cbx}. The operations
�9 , / easily extend to subsets, for example AB is
~J{ab : a cA,b CB}. In particular a(bc) and (ab)c are
subsets of X. Note that A/B = C if and only if A = BC.
The most common example of this situation is when
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2 BRYANT
X is a real vector space and
a.b = {%a + (I-%) b : O < X < I}, and in that case
a/b = {%a + (l-%)b : % > I}. The axioms which we con-
sider have a strong geometric motiviatlon based upon
this example.
DEFINITION. A pair (X,.) is a Convexity Space (or
Join-structure) if �9 is a join on X satisfying
(i) ab # r
(2) a/b ~ r
(3) aa = a = a/a
(4) ab = ba
(5) (ab)c = a(bc)
(6) a/b = c/d => ad = bc
for all a,b,c,dcX.
Some of the consequences of these axioms are
studied in E2-~ and E6~. Other similar axiomatic
approaches can be found, for example, in ~-3] - E53.
THEOREM. If �9 is a J@in on x satisfying
(I) a/b # r
(II) aa = a = a/a
(III) (ab)cCa(bc)
(IV) a/b = c/d => ad = bc
for all a,b,c,d ~X, then (X,-) is a Convexity Space.
Proof. Assume that (X,') satisfies I - IV. We show
that properties i, 4 and 5 hold and it will then follow
that (X,') is a Convexity Space.
(i) ah + r
For each a,bCX we have, by I, a/b ~ @. Thus
a/b = a/b and by IV ab = ba. Hence ab~ab ~ ba ~ r
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BRYANT 3
as required.
(4) ab = ba:
We show that for any a,b abcba and then 4 will
follow by symmetry. So let c Cab and by 1 above we
may choose d Ccb. Then b C d/c ~c/a and so by IV
da = cc. Thus by II and III
c = cc = daC(cb)a ~c(ba)
and so c cc(ba), c/c = ba and c cba as required.
(5) (ab)c = a(bc):
By III and 4 above we have
(ab)c ~a(bc) = (bc)acb(ca) = (ca)b~c(ab) = (ab)c
and the result follows. (Note that this is an example
of the construction in (iii) of ~I~.)
Hence (X,.) is a Convexity Space as claimed.
We conclude by giving examples to show that the
five properties I, IIi (aa = a), IIii (a/a = a), III
and IV are independent. The examples chosen are not
the simplest but they do illustrate why the particular
property fails.
EXAMPLE I. If X = {x ~ R : O $ x $ I} and �9
denotes the usual Vector Space join defined above,
then (X,') satisfies II, III and IV but 0/I = @.
EXAMPLE lli. If X = R and joins are defined by
a.b = {xC R : either x < min {a,b} or x -- b},
then clearly (X,-) satisfies I and IIii. Also
(ab)c = {x : x < min {y,c} some y Cab, or x -- c}
= {x : x < min {a,b,c}, x < min {b,c} or x = c}
= {x : x < min {b,c} or x = c}
= bcC a(bc)
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4 BRYANT
and so III holds. Furthermore ad = bc for all
a,b,c,dcX and so IV trivially holds. However
aa = {xCR : x ~ a} and so property lli fails.
EXAMPLE llii. If X = {O,I} and �9 is defined by
O.O = O, I.i = i, O.i = i.O = {O,I}, then (X,') satis-
fies I, lli, III and IV. However property llii fails
since I/i = {0,i}.
EXAMPLE III. Let X = R and let
a.b = f a if a = b
L R\{a,b} if a + b
Then in this case a/b = ab for each a,b and it is easy
to check that (X,') satisfies I, II and IV. For
example if ad + bc then exactly three of a,b,c and d
must be the same point a~d then a/b = ab # cd = c/d.
However property III fails for if a + b then
(ab)b = (R\{a,b})b = R~b
and a(bb) = ab = R\{a,b}
whence (ab)b~a(bb).
EXAMPLE IV. If X = R2\(0,0) and the join is
defined as the usual Vector Space join excluding (0,0),
then (X,') satisfies I, II and III. However
(2,2) C(O,I)/ (-2,O) rl (I,O)/ (0,-2)
and (O,i) . (O,-2)~(-2,O) . (I,O) = ~.
Thus property IV fails in this case.
We have thus shown that the properties I - IV
form independent axioms for a Convexity Space or Join-
structure. Sincere thanks are due to Roger Webster of
Sheffield for his original work concerning the axioms.
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BRYANT 5
REFERENCES
I. V. W. Bryant: Reducing classical axioms, Math. Gazette, 391 (1971) 38-40.
2. V. W. Bryant and R. J. Webster: Generalizations of the theorems of Radon, Helly and Caratheodory, Monatsh. Math., 73 (1969) 309-315.
3. J. W. Ellis: A general set-separation theorem, Duke Math. J., 19 (1952) 417-421.
4. A. Ghika: Separarea multimilor convexe in spatii lineate non- vectoriale, Acad. R.P. Romine Bul. Sti. Sect. Sti. Mat. Fiz., 7 (1955) 287-296.
5. V. Havel: Join systems and closure spaces, Comment Math. Univ. Corolinae, 7 (1966) 335-341.
6. W. Prenowitz: A contemporary approach to classical geometry, Amer. Math. Monthly, 68 (1961) Appendix 1-67.
Department of Pure Mathematics, The ~liversity of Sheffield, Sheffield $3 7RH, England.
(Eingegaugen am ~5. Februar 1974)
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