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TOPOLOGYPROCEEDINGSVolume 29, No.2, 2005

Pages 539-558

HOFMANN-MISLOVE POSETS

MARTIN MARIA KOVAR∗

Abstract. In this paper we attempt to find and investigatethe most general class of posets which satisfy a properly gen-eralized version of the Hofmann-Mislove theorem. For thatpurpose, we generalize and study some notions (like compact-ness, the Scott topology, Scott open filters, prime elements,the spectrum etc.), and adjust them for use in general posetsinstead of frames. Then we characterize the posets satisfyingthe Hofmann-Mislove theorem by the relationship betweenthe generalized Scott closed prime subsets and the general-ized prime elements of the poset. The theory becomes classicfor distributive lattices. The topologies induced on the gen-eralized spectra in general need not be sober.

1. Introduction and Terminology

The Hofmann-Mislove theorem is one of the most often appliedresults in theoretical computer science and computer sciencemotivated topology. It says that there is a 1-1 correspondencebetween Scott open filters of a frame and compact saturated sets ofits abstract points which can be naturally represented as the primeelements of the frame. Yet recent developments in the topic indicatethat the frame structure of a poset or the sobriety of a topologi-cal space impose an unwanted limitation on the classic theory in

2000 Mathematics Subject Classification. 54F05, 54H99, 6B30.Key words and phrases. Posets, generalized Scott topology, Scott open fil-

ters, (filtered) compactness, saturated sets, prime elements and prime subsets.∗The author acknowledges support from Grant 201/03/0933 of the Grant

Agency of the Czech Republic and from the Research Intention MSM0021630503 of the Ministry of Education of the Czech Republic.

539

540 MARTIN MARIA KOVAR

some cases. The author can maintain this assertion by his personalexperience with investigating the properties of the de Groot dual,or in solving the question of D. E. Cameron, whether every com-pact topology is contained in some maximal compact topology [2].In both cases, some modification of the Hofmann-Mislove theoremis useful. However, the author believes that the utility of the pre-sented generalization is not limited only to these two topics. Someother applications, perhaps more close to theoretical computer sci-ence, might be found in the future. In this paper we attempt toreach the boundaries of possible generalizations which, however,still leave the core and the main principles of the Hofmann-Mislovetheorem untouched. The presented generalized version reduces tothe classic theory if the considered poset is a distributive lattice.The topological formulation of the Hofmann-Mislove theorem willbe studied in more detail in a separate paper.

In this section we explain most of the used terminology, except forsome primitives and notions that we touch only marginally. Thesenotions are not essential for understanding the paper. However, fora thorough introduction and detailed explanation of the topics thereader is referred to the books and monographs [3], [7] and [8].

Let P be a set, ≤ a reflexive and transitive, but not necessarilyantisymmetric binary relation on P . Then we say that ≤ is a pre-order on P and (P,≤) is a preordered set. For any subset A of apreordered set (P,≤) we denote ↑A = {x | x > y for some y ∈ A}and ↓A = {x | x 6 y for some y ∈ A}. An important example of apreordered set is given by the preorder of specialization of a topolog-ical space (X, τ), which is defined by x ≤ y if and only if x ∈ cl {y}.This preorder is a partial order in the usual sense if and only if thespace (X, τ) is T0. For any x ∈ X it is obvious that ↓ {x} = cl {x}.A set is said to be saturated in (X, τ) if it is an intersection of opensets. One can easily verify that a set A ⊆ X is saturated in (X, τ)if and only if A = ↑A, that is, if and only if A is an upper set withrespect to the preorder of specialization of (X, τ). Thus for everyset B ⊆ X , the set ↑B is called the saturation of B. Compact-ness is understood without any separation axiom. The family of allcompact saturated sets in (X, τ) is a closed base for a topology τd,which is called de Groot dual of the original topology τ . A topolog-ical space is said to be sober if it is T0 and every irreducible closedset is a closure of a (unique) singleton.

HOFMANN-MISLOVE POSETS 541

Let ψ be a family of sets. We say that ψ has the finite intersectionproperty, or briefly, that ψ has f.i.p., if for every P1, P2, . . . , Pk ∈ ψit follows P1 ∩P2 ∩ · · ·∩Pk 6= ∅. Let (X, τ) be a topological space,and let Φ,Ψ ⊆ 2X . Recall that a set S ⊆ X is compact with re-spect (or relative) to the family Φ, if for every subfamily ϕ ⊆ Φsuch that {S} ∪ ϕ has f.i.p. it follows S ∩ (

⋂ϕ) 6= ∅. We say that

the family Φ is Ψ-up-conservative (Ψ-down-conservative, respec-tively) if for every A ∈ Φ and B ∈ Ψ it follows ↑ (A ∩ B) ∈ Φ(↓ (A ∩ B) ∈ Φ, respectively). We say that Ψ is upper-closed(lower-closed, respectively) if for every A ∈ Ψ it follows A =↑ A(A =↓ A, respectively). The family Ψ is said to be up-compact(down-compact, respectively) if every A ∈ Ψ is compact with re-spect to the family {↑ {x} | x ∈ X} ({↓ {x} | x ∈ X}, respectively).The family Ψ is said to be up-complete (down-complete, respec-tively) if {↑ {x} | x ∈ X} ⊆ Ψ ({↓ {x} | x ∈ X} ⊆ Ψ, respec-tively).

Let (X,≤) be a partially ordered set, or briefly, a poset. If (X,≤)has, in addition, finite meets, then any element p ∈ X is said tobe prime if x ∧ y ≤ p implies x ≤ p or y ≤ p for every x, y ∈ X .The set P of all prime elements of (X,≤) is called the spectrumof (X,≤). We say that the poset (X,≤) is directed complete, orDCPO, if every directed subset of X has a least upper bound – asupremum. A subset U ⊆ X is said to be Scott open, if U = ↑U andwhenever D ⊆ X is a directed set with supD ∈ U , then U∩D 6= ∅.One can easily check that the Scott open sets of a DCPO form atopology. This topology we call the Scott topology. Thus a setA ⊆ X is closed in the Scott topology if and only if A = ↓A and ifD ⊆ A is directed, then supD ∈ A. It follows from Zorn’s Lemmathat in a DCPO, every element of a Scott closed subset S is belowsome maximal element of S. It is easy to see that the closure of asingleton {x} in the Scott topology is ↓ {x}, thus the original order≤ of X can be recovered from the Scott topology as the preorderof specialization.

Let us describe some other important topologies on posets. Theupper topology [3], which is also referred as the weak topology [7]or the lower interval topology [6] has the collection of all principallower sets ↓ {x}, where x ∈ X , as the subbase for closed sets. Thepreorder of specialization of the lower interval topology coincideswith the original order of (X ≤). Hence, the saturation of a subset


A ⊆ X with respect to this topology is ↑A. Similarly, the lowertopology, also referred as the weakd topology [7] or the upper intervaltopology [6], arises from a subbase for closed sets which consistsof all principal upper sets ↑ {x}, where x ∈ X . Note that theweakd topology is not the de Groot dual of the weak topology ingeneral; the weakd topology is the weak topology with respect to theinverse partial order. The preorder of specialization of the upperinterval topology is a binary relation inverse to the original orderof (X,≤). Consequently, the saturation of a subset A ⊆ X withrespect to this topology is ↓A. The topology on the spectrum Pof a directed complete ∧-semilattice (X,≤), induced by the upperinterval topology, is called the hull-kernel topology [7].

Let (X,≤) be a poset. The set F ⊆ X is said to be filtered, ifevery finite subset of F has a lower bound in F . Since the emptyset is included, it has a lower bound in F which is, therefore, non-empty. If, in addition, F = ↑F then F is called a filter on (X,≤).In this setting, a filter base in a set X can be defined as a filteredset ϕ ⊆ 2X in the poset (2X ,⊆), such that ∅ /∈ ϕ.

2. Filtered Compactness

and the generalized Scott topology

We will start with an example which illustrates the relationshipbetween the Scott topology and compactness in terms of de Grootdual. In [5] the author proved that for a given topological space(X, τ) with the family of compact saturated sets K it holds τ = τdd

if and only if (X, τ) has an up-compact, K-down-conservative closedsubbase. We need this result for the example.

Example 2.1. Let (X,≤) be a frame, ω be the upper-intervaltopology on X , σ be the Scott topology on X . We leave to thereader to show that the compact saturated sets in (X,ω) are exactlythe Scott closed sets, so σ = ωd (the reader can, e.g., adjust theproof of Proposition 2.3). For every a ∈ X , the principal filter ↑ {a}is up-compact and, for every Scott closed K ⊆ X , ↑(↑ {a}∩K) = ∅or ↑(↑ {a} ∩K) = ↑ {a}, so the family of principal filters extendedby the empty set is down-conservative with respect to the family ofthe Scott closed sets. By the previously mentioned result, ω = ωdd.Hence, ω and σ are the de Groot duals of each other. �


However, this relation between the upper-interval topology andthe Scott topology need not remain true in any of the both direc-tions if we replace the frames by more general posets. This factnaturally leads to the following, slightly adjusted notion of com-pactness.

Definition 2.1. Let X be a set, Φ ⊆ 2X . We say that K ⊆ Xis filtered compact with respect to the family Φ if K ∩ (

⋂ϕ) 6= ∅

for every filter base ϕ ⊆ Φ such that each of its elements meet K.In a poset (X,≤) we say that K ⊆ X is up-filtered compact, ifit is filtered compact with respect to the family {↑ {x} | x ∈ X} ofprincipal upper sets.

Throughout the paper we work especially with up-filtered com-pactness of general posets, but we may be interested what thisnotion means in terms of topological spaces and how it differs fromusual compactness. The following analogue of Alexander’s subbasetheorem describes filtered compactness in terms of convergence ofmore general families than filter bases consisting of members of thegiven closed subbase.

Proposition 2.1. Let (X, τ) be a topological space, C the family ofall closed sets, C0 ⊆ C a closed subbase. The following statementsare equivalent for a subset K ⊆ X:

(i) K is filtered compact with respect to C0.(ii) For every filter base ϕ ⊆ C whose every element meets K

such that ϕ ∩ C0 is a filter base, K ∩ (⋂ϕ) 6= ∅.

(iii) For every family ϕ ⊆ C such that ϕ ∪ {K} has f.i.p. andϕ ∩ C0 is a filter base, K ∩ (

⋂ϕ) 6= ∅.

Proof. It is clear that (iii) → (ii) → (i). Suppose (i). We say that afamily ϕ ⊆ C has property P if ϕ∪{K} has f.i.p. and ϕ∩C0 is a filterbase. Let L be a chain of closed families having property P , linearlyordered by set inclusion. It is easy to check that

⋃L again has P .

Let ϕ ⊆ C be a family with P . By Zorn’s Lemma, ϕ is containedin some maximal family having P , say ψ. We put ψ0 = ψ ∩ C0. Itfollows from (i) that there exists some p ∈ K ∩ (

⋂ψ0). Suppose

that p /∈⋂ϕ. Then there exists F ∈ ϕ such that p /∈ F . But

F ∈ C, so there exists a set A, for every α ∈ A a finite set Iα,and for every i ∈ Iα a closed set Ci ∈ C0, such that F =

⋂α∈A Fα,

where Fα =⋃

i∈Iα

Ci. There exists some β ∈ A such that p /∈ Fβ.


We have F ⊆ Fβ and ψ∪{K} has f.i.p., so ∅ 6= K∩F∩P1∩P2∩· · ·∩Pk ⊆ K∩Fβ ∩P1∩P2∩· · ·∩Pk for every P1, P2, . . . , Pk ∈ ψ. Hence,there exists m ∈ Iβ such that Cm has the same property as Fβ , i.e.,for every P1, P2, . . . , Pk ∈ ψ we haveK∩Cm∩P1∩P2∩· · ·∩Pk 6= ∅.

We put ψ′ = ψ ∪ {Cm} ∪ {Cm ∩ (⋂k

j=1 Pj) | Pj ∈ ψ0, j = 1, . . . , k}.

Then ψ′ has property P , so from the maximality of ψ it followsψ′ = ψ and we have Cm ∈ ψ ∩ C0. Then p ∈ Cm ⊆ Fβ , which is acontradiction. Hence, p ∈ K ∩ (

⋂ϕ), which yields (iii). �

Corollary 2.1. Let (X, τ) be a topological space, τ0 ⊆ τ an opensubbase of τ . The following statements are equivalent for a subsetK ⊆ X:

(i) K is filtered compact with respect to C0 = {XrU | U ∈ τ0}.(ii) For every directed open cover O ⊆ τ such that O ∩ τ0 is

directed, there exists U ∈ O containing K.(iii) For every open cover O ⊆ τ such that O ∩ τ0 is directed,

there exists finite O′ ⊆ O covering K.

In the following example we will show that in general, com-pactness and filtered compactness are different properties. Notethat the construction of the topological space is due to B. Burdick[1], who used it as an example of a space whose iterations of thede Groot dual (including the original topology) can generate fourdifferent topologies.

Example 2.2. Let (X, τ) be the first uncountable ordinal X = ω1

equipped with the topology τ ={〈0, α)rF | 06α6ω1, F is finite}.Then any closed set in (X, τ) has the form C = 〈α, ω1) ∪ F , where0 6 α 6 ω1 and F is finite. It is easy to see that (X, τ) is a T1

space. Suppose that C is a non-empty closed set which is not asingleton. If α = ω1, then C = F has at least two elements and itis easy to decompose it into two strictly smaller non-empty closedsets. If α < ω1, we have C = {α} ∪ 〈α + 1, ω1)∪ F . In any case, Cis not irreducible. Then (X, τ) is sober. We leave to the reader tocheck that τd is the cocountable topology.

Now we will continue directly with the previously constructedspace (X, τ), but alternatively, instead of (X, τ) one can use alsoany sober space whose de Groot dual is not compact. Since the fam-ily Φ of all compact saturated sets is a closed base for (X, τd), byAlexander’s subbase theorem X is not compact with respect to Φ.


However, in a sober space, any filter base consisting of non-emptycompact saturated sets has a non-empty intersection (see [4], Corol-lary 2), so X is filtered compact with respect to Φ. �

Note that if the closed subbase C0 of (X, τ) is closed under bi-nary intersections, filtered compactness with respect to C0 coincideswith compactness. If the poset (X,≤) has binary joins, the fam-ily {↑{x} | x ∈ X} is closed under binary intersections. Hence,in this case, up-filtered compact means the same as compact withrespect to the upper interval topology. Similarly as compactness,up-filtered compactness of a set is equivalent to up-filtered com-pactness of its saturation.

Proposition 2.2. Let (X,≤) be a poset. ThenK ⊆ X is up-filteredcompact if and only if ↓K is up-filtered compact.

Proof. Let U ⊆ X , U = ↑U . It is easy to observe that K ∩ U 6= ∅if and only if ↓K ∩ U 6= ∅. Applying this observation to U = ↑{a}or U =

⋂ϕ, where ϕ = {↑{a} | a ∈ A} is a filter base from

the definition of up-filtered compactness (cf. Definition 2.1 and itsnotation), one can complete the proof. �

We would like to have a similar relationship between the upperinterval topology and the Scott topology as it is demonstrated forframes in Example 2.1. In a DCPO, as one can prove, the Scottclosed sets are exactly the up-filtered compact saturated sets withrespect to the upper interval topology. However, how to extend theScott topology to posets which are not directed complete? Thereare, at least, two outmost possibilities. For instance, one can definethat a lower set is Scott closed if its each directed subset has asupremum, which is contained in the lower set. In this case we canget very few Scott closed sets. Another, rather extreme possibility isto define that the lower set contains suprema of its directed subsetsonly if the suprema exist. This definition may generate too largea family of Scott closed sets. In DCPO’s both cases coincide withthe original definition of the Scott closed set, but, unfortunately, forgeneral posets they need not work properly. We can demonstrateit by an example.

Example 2.3. Let X = R r {0, 2} and let ≤ be the natural lin-ear order of the real numbers. We put A = {x | x ∈ X, x ≤ −1},B = {x | x ∈ X, x ≤ 1} and C = {x | x ∈ X, x ≤ 2}.


Then only the set A matches the first, the strongest possibility.All the three sets A, B, C match the second, the weakest possibledefinition of a Scott closed set. The upper interval topology on(X,≤) is the family τ = {∅, X}∪ {(−∞, a)∩X | a ∈ X}. Clearly,all the three sets A, B, C are saturated. However, the sets A, B arecompact in this topology, but C is not compact. Hence, choosingthe first possibility, there would be more compact saturated setsthan the Scott closed sets, while choosing the second possibilitywould cause too many Scott closed sets and some of them wouldnot be compact. �

In the previous example, a compromise solution which works wellis represented by the set B. In general, it is given by the followingdefinition.

Definition 2.2. Let (X,≤) be a poset. We say that A ⊆ X is aScott closed basic set, if A = ↓A and each directed D ⊆ A has anupper bound in A. A set B ⊆ X is said to be a Scott open basicset, if X r B is a Scott closed basic set.

For our convenience, we will use the shortcuts ‘SCB set’ for ‘Scottclosed basic set’ and ‘SOB set’ for ‘Scott open basic set’. We leaveto the reader to check that the family of the Scott closed basic setsis closed under finite unions. However, in a general case it neednot be closed under intersections, as we can see from the followingexample. Hence, the Scott open basic sets form a base for the opensets of some topology, but it itself need not be a topology in general.

Example 2.4. Let (N,≤) be the set of natural numbers with theirnatural order, and let a, b /∈ N, a 6= b. We put X = N ∪ {a, b}. Forany x, y ∈ X we put x 6 y if and only if any of the following casesis fulfilled:

(i) x, y ∈ N, x < y,(ii) x ∈ N, y ∈ {a, b},(iii) x = y.

We leave to the reader to check that 6 is a reflexive, antisymmetricand transitive relation. Let A = N∪ {a}, B = N∪ {b}. Then A, Bare SCB sets, but in A ∩ B its directed subset N ⊆ A ∩ B has noupper bound, so A ∩B is not an SCB set. �

Definition 2.3. The topology generated by the family of SOB setswe call the generalized Scott topology.


Proposition 2.3. Let (X,≤) be a poset. Then K ⊆ X is an SCBset if and only if K is saturated in the upper interval topology andup-filtered compact.

Proof. Let K ⊆ X be an SCB set. The specialization preorderof the upper interval topology is the opposite of the given partialorder ≤, so K is saturated (in the upper interval topology). Letϕ = {↑ {a} | a ∈ A} be a filter base such that K ∩ ↑ {a} 6= ∅ forevery a ∈ A. Since ϕ is a filter base then if a, b ∈ A there existsc ∈ A such that ↑ {c} ⊆ ↑{a} ∩ {b}, that is, c ≥ a, b. So A isdirected. Further, if a ∈ A, then there is some x ∈ K ∩ ↑{a}. Itfollows that a ≤ x and since K is a lower set, we have a ∈ K. HenceA ⊆ K. Then A has an upper bound u ∈ K since K is an SCBset. But then u ∈ K ∩ (

⋂a∈A ↑ {a}). It means that K is up-filtered

compact.Conversely, let K ⊆ X be up-filtered compact and saturated in

the upper interval topology. Then there exists F ⊆ X such thatK =

⋂a∈F (X r ↑ {a}) and, consequently, K is a lower set. Let

A ⊆ K be directed. Then Φ = {↑ {a} | a ∈ A} is a closed filterbase and all its elements clearly meet K. Since K is up-filteredcompact, there exists u ∈ K ∩ (

⋂a∈A ↑ {a}). Then u ≥ a for every

a ∈ A, so u is an upper bound of A which is contained in K. Hence,K is an SCB set. �

One can state a natural question whether the filtered version ofthe de Groot dual applied on the generalized Scott topology alwaysyields back to the original upper interval topology of the posetsimilarly as we described in Example 2.1 for frames. The authorso far has no definitive answer for that simple question, althoughthe expected answer is ‘no’. The general iteration properties of thismodified de Groot dual still remain open, too.

3. Hofmann-Mislove posets

The Hofmann-Mislove Theorem says that there is a 1-1 corre-spondence between Scott open filters of a frame and compact sat-urated sets of its abstract points which can be naturally repre-sented as the prime elements of the frame. The set of the primeelements is known as the spectrum of the frame. However, ifwe have a more general poset than a frame, this correspondenceeither need not work at all or, at least, not so straightforward.


In this chapter we attempt to find the most general class of posetsthat satisfy a proper generalization of the Hofmann-Mislove Theo-rem. To be able to do this, we need to adjust some notions whichare very simple and clearly understood in frames but are rathermore complicated in a general setting. In the following definitionwe modify and extend the notion of a prime element to be relevantalso for those posets which do not necessarily have the finite meets.

Definition 3.1. Let (X,≤) be a poset, L ⊆ X. We say that L isprime if ↓L 6= X and for every a, b ∈ X

↓ {a} ∩ ↓{b} ⊆ ↓L⇒ (a ∈ ↓L) ∨ (b ∈ ↓L).

It can be easily seen that if (X,≤) has finite meets, any elementp ∈ X is prime if and only if the singleton {p} is prime as a set.Hence, we can extend the notion of a prime element also to thoseposets which do not necessarily have finite meets. Thus in thefollowing text we mean that an element p of a poset (X,≤) is primeif and only if {p} is prime in the sense of the previous definition.As the following proposition shows, the notions of a prime set andof a filter are dual.

Proposition 3.1. Let (X,≤) be a poset. Then L ⊆ X is prime ifand only if F = X r ↓L is a filter.

Proof. Let L ⊆ X be prime. Then F = Xr↓L is a nonempty upperset. Let a, b ∈ F . Then a, b /∈ ↓L, which implies that there is somec ∈ ↓{a} ∩ ↓{b} such that c /∈ ↓L, i.e. c ∈ F . Conversely, let F bea filter. Then ↓L = X r F 6= X . Suppose that ↓ {a} ∩ ↓{b} ⊆ ↓Lfor some a, b ∈ X . Then a, b /∈ ↓L implies a, b ∈ F , which meansthat there is some c ≤ a, c ≤ b, c ∈ F . Then c ∈ ↓{a} ∩ ↓ {b}, butc /∈ ↓L, which is a contradiction. �

In the direct proof of the topological formulation of the Hofmann-Mislove Theorem (see, e.g. [4]), the sobriety of the topological spaceis needed to ensure that if an open set contains an intersection of aScott open filter, then this open set is an element of the filter. Wemay say that such a filter is “wide” enough. If a topological spaceis not sober, its topology can have Scott-open filters which are notwide in this sense, but, on the other hand, the Scott open filtersgenerated by compact sets are always wide. We want to modelthis situation in a poset equipped with the upper-interval topology.


However, all the elements of a poset need not necessarily correspondto the points of a certain topological space in the analogy thatwe want to model. It will be more convenient to relativize the“wideness” of a filter to subsets of posets and then study whichsubsets have the desired properties, whatever they are.

Definition 3.2. Let (X,≤) be a poset, P ⊆ X. Denote ψ(x) =P r ↑{x} for every x ∈ X. We say that non-empty set F ⊆ X iswide relative to P , if for every a ∈ X,

⋂x∈F ψ(x) ⊆ ψ(a) ⇒ a ∈ F .

Since we will often work with the prime sets rather than withfilters, according to Proposition 3.1 we need a notion dual to relativewideness. In particular, a Scott closed prime set should have thatproperty if and only if its complement is a relatively wide Scottopen filter. This is a motivation for the next definition and theconsecutive proposition, which only shows that the new notion hasthe expected and desired properties. There is only one exception– the notion of relative narrowness defined below make sense evenfor the case K = X . On the other hand, the empty set always(and, unfortunately, independently on P ) satisfies the condition ofrelative wideness.

Definition 3.3. Let (X,≤) be a poset, P ⊆ X. We say that K ⊆ Xis narrow relative to P if K ⊆ ↓(P ∩K).

Proposition 3.2. Let (X,≤) be a poset, K ( X a lower set. ThenK is narrow relative to P if and only if F = XrK is wide relativeto P .

Proof. Let K ( X narrow relative to P , K = ↓K. Then ∅ 6=F = X r K is an upper set and

⋂x∈F ψ(x) =

⋂x∈F (P r ↑{x}) =

P r⋃

x∈F ↑{x} = P rF = P ∩K. Suppose that⋂

x∈F ψ(x) ⊆ ψ(a)for some a ∈ X . Then P ∩ K ⊆ P r ↑{a}, which means thatP ∩ K ∩ ↑{a} = ∅. Then a /∈ ↓(P ∩ K), and since K is narrowrelative to P , a /∈ K. It follows a ∈ F , which yields that F is widerelative to P . Conversely, suppose that F is wide relative to P . Leta ∈ K = X r F . Then

⋂x∈F ψ(x) = P ∩ K * ψ(a) = P r ↑{a}.

Then P ∩ K ∩ ↑{a} 6= ∅, so there exists some t ∈ P ∩ K ∩ ↑{a}.Then a ≤ t and t ∈ P ∩ K, which gives a ∈ ↓(P ∩ K). Hence,K ⊆ ↓(P ∩K). �

Another, also useful characterization of relative narrowness isgiven by the following proposition.


Proposition 3.3. Let (X,≤) be a poset, K ⊆ X a lower set. ThenK is narrow relative to P if and only if there exists L ⊆ P suchthat K = ↓L.

Proof. Let K ⊆ X be narrow relative to P . Then K ⊆ ↓(P ∩K).We put L = P ∩ K and since K is a lower set, we have K = ↓L.Conversely, suppose that K = ↓L, where L ⊆ P . Let x ∈ K.Then there exists some t ∈ L ⊆ P ∩ K such that x ≤ t. Thenx ∈ ↓(P ∩ K), which gives K ⊆ ↓(P ∩ K). Hence, K is narrowrelative to P . �

Now, we are ready to say more precisely what we mean by theanalogy with frames or sober topological spaces that we want tomodel for a certain class of posets. The desired situation is de-scribed by the conditions (i) and (ii) of the following propositionequivalently in terms of the SOB filters and SCB prime sets.

Proposition 3.4. Let (X,≤) be a poset, ω the upper intervaltopology on X, ωP the induced topology on P ⊆ X. The follow-ing conditions (i) and (ii) are equivalent:

(i) There exists P ⊆ X such that:(1) For every SOB filter F ⊆ X, if we denote ψ(x) =

Pr↑ {x} and L =⋂

a∈F ψ(a), the set L is up-filtered com-pact, saturated in (P, ωP ) and F = {x | x ∈ X,L ⊆ ψ(x)}.

(2) For every up-filtered compact and saturated L ⊆ P in(P, ωP ), the set F = {x | x ∈ X,L ⊆ ψ(x)} is a SOBfilter.

(ii) There exists P ⊆ X such that:(1) For every SCB prime set K ⊆ X, the set L = P ∩K is

up-filtered compact, saturated in (P, ωP ) and K = ↓L.(2) For every up-filtered compact and saturated L ⊆ P in

(P, ωP ), the set ↓L is an SCB prime set.

Proof. As it follows from Proposition 3.1 and Definition 2.2,F ⊆ X is an SOB filter if and only if K = X r F is an SCBprime set. Further, we have

⋂a∈F ψ(a) =

⋂a∈F (P r ↑ {x}) =

Pr⋃

a∈F ↑ {a}=PrF =P∩K, andXr↓L={x | x ∈ X, x /∈ ↓L} ={x | x ∈ X,L∩ ↑{x} = ∅} = {x | x ∈ X,L ⊆ ψ(x)}. Now it isclear that (ii) is only a reformulation of (i). �


Definition 3.4. Let (X,≤) be a poset. We say that (X,≤) isHofmann-Mislove, if (X,≤) satisfies any of the conditions (i) or(ii) of Proposition 3.4. The set P ⊆ X from (i) or (ii) we call ageneralized spectrum of (X,≤); the topology ωP we call the gener-alized hull-kernel topology on P .

The natural question that we immediately have to ask justafter the definition is, which posets are Hofmann-Mislove and howmany generalized spectra such a poset can have. The next propo-sition shows, as one can expect, that the generalized spectrum isdetermined uniquely.

Proposition 3.5. Let (X,≤) be a Hofmann-Mislove poset, S ⊆X any generalized spectrum. Then S = {p | p ∈ X, p is prime} ={m | m is a maximal element of an SCB prime subset of X}.

Proof. Let M = {m |m is a maximal element of an SCB primeset}, P = {p | p ∈ X, p is prime}. Let p ∈ P be a prime element.Then ↓ {p} is an SCB prime set and p is its maximal element. ThusP ⊆M . Let m ∈M and let K ⊆ X be an SCB prime set such thatm ∈ K is a maximal element. ThenK = ↓(K∩S), som ∈ ↓(K∩S).Then there exists t ∈ K ∩ S with m ≤ t. But m is maximal in K,so m = t ∈ S. Hence, M ⊆ S. Let s ∈ S. Then {s} is clearly up-filtered compact, as well as ↓ {s}. The set L = S∩↓ {s} is saturatedin (S, ωS) and ↓L = ↓ {s}. Hence, L is up-filtered compact. Then↓L is an SCB prime set, which means, in particular, that s is prime.Therefore, S ⊆ P . Now we have P ⊆M ⊆ S ⊆ P , which completesthe proof. �

The rest of the section will be devoted to simplifying the condi-tion of being Hofmann-Mislove.

Proposition 3.6. A poset (X,≤) is Hofmann-Mislove if and onlyif there exists P ⊆ X such that the following statements are fulfilled:

(1) Every SCB prime set is narrow relative to P .(2) Every L ⊆ P such that ↓L is an SCB set is prime.

Proof. We will show that the conditions (1) and (2) are equivalentto the corresponding conditions of Proposition 3.4. It is clear thatthe condition (1) of (ii) in Proposition 3.4 implies (1). Conversely,suppose (1). Then, for every SCB prime set K, it holds K ⊆↓(P ∩ K). Since K = ↓K, we have K = ↓L, where L = P ∩ K.


It follows from Proposition 2.2 that L is up-filtered compact and,clearly, L is saturated in (P, ωP ). Hence, (1) and the condition (1)of (ii) in Proposition 3.4 are equivalent.

Now, suppose (2) of (ii) in Proposition 3.4. Let L ⊆ P suchthat ↓L is an SCB set. Then, by Proposition 2.3, ↓L is up-filteredcompact and, of course, saturated in the upper-interval topology.We put M = P ∩ ↓L. Then L ⊆ M ⊆ ↓L, so ↓M = ↓L. Then, byProposition 2.2, M is up-filtered compact. Since ↓L is saturatedin (X,ω), M is saturated in (P, ωP ). By (2) of (ii) in Proposi-tion 3.4, ↓M = ↓L is prime. By Definition 3.1, L is prime. Hence,(2) is fulfilled. Conversely, suppose (2). Let L ⊆ P be up-filteredcompact and saturated in (P, ωP ). Then ↓L is also up-filtered com-pact by Proposition 2.2 and saturated in (X,ω) as a lower set. ByProposition 2.3, ↓L is an SCB set. It follows from (2) that ↓L isprime, so the condition (2) of (ii) in Proposition 3.4 is fulfilled. Thiscompletes the proof. �

Theorem 3.1. A poset (X,≤) is Hofmann-Mislove if and onlyif there exists P ⊆ X such that for every SCB set K ⊆ X, thefollowing statements are equivalent:

(i) K is prime.(ii) K is narrow relative to P .

Proof. Suppose that (X,≤) is Hofmann-Mislove and let P be itsgeneralized spectrum. Let K ⊆ X be an SCB set. If K is prime,then by Proposition 3.6 K is narrow relative to P . Conversely, ifK is narrow relative to P , by Proposition 3.3 K = ↓L for someL ⊆ P . By Proposition 3.6, L is prime and so K is also prime.Therefore, the conditions (i) and (ii) are equivalent.

Conversely, suppose that there exists P ⊆ X such that for everySCB set the conditions (i) and (ii) are equivalent. Let K ⊆ Xbe an SCB prime set. By the implication (i) → (ii), K is narrowrelative to P , so the condition (1) of Proposition 3.6 is fulfilled.Now, let L ⊆ P be such that K = ↓L is an SCB set. Then K isnarrow relative to P by Proposition 3.3. Therefore, K is prime bythe implication (ii) → (i). It follows just from Definition 3.1 thatL is prime. Hence, the condition (2) of Proposition 3.6 holds. ByProposition 3.6, (X,≤) is Hofmann-Mislove. �


Combining the previous theorem with Proposition 3.5, we havethe following corollary.

Corollary 3.1. Let (X,≤) be a poset, P ⊆ X the set of primeelements. Then (X,≤) is Hofmann-Mislove if and only if for everySCB set K ⊆ X, the following statements are equivalent:

(i) K is prime.(ii) K is narrow relative to P (i.e., K = ↓L for some L ⊆ P ).

The previous result can also be reformulated without the explicituse of the generalized spectrum, as we can see from the followingtheorem.

Theorem 3.2. A poset (X,≤) is Hofmann-Mislove if and only iffor every SCB set K ⊆ X the following statements are equivalent:

(i) K is prime.(ii) Every maximal element of K is prime.

Proof. Let P be the set of prime elements of (X,≤). Suppose that(X,≤) is Hofmann-Mislove. Let K be an SCB set. If K is prime,by Corollary 3.1 K = ↓L, where L ⊆ P . Let m ∈ K be a maximalelement of K. Then there exists some p ∈ L such that m ≤ p andfrom maximality we have p = m. Hence, every maximal element ofK is prime. Conversely, suppose that every maximal element of Kis prime. Let M ⊆ K be the set of maximal elements of K. ThenM ⊆ P . Since K is an SCB set, it follows from Zorn’s Lemma thatevery element x ∈ K is comparable with some maximal elementm ∈ M – we have x ≤ m. Then K = ↓M , so by Proposition 3.3K is narrow relative to P . It follows from Corollary 3.1 that K isprime. Hence, the conditions (i) and (ii) are equivalent.

On the other hand, suppose that the conditions (i) and (ii) areequivalent for every SCB set K ⊆ X . Let K be prime. Then, bythe implication (i) → (ii) it follows that every maximal element ofK is prime. If M ⊆ K is the set of maximal elements of K, thenK = ↓M and M ⊆ P . By Proposition 3.3, K is narrow relative toP . Conversely, let K be narrow relative to P and let m ∈ K be amaximal element. We have K = ↓L for some L ⊆ P , which yieldsm ∈ L because of maximality of m. Then every maximal elementof K is prime. It follows from the implication (ii) → (i) that K isprime. By Corollary 3.1, (X,≤) is Hofmann-Mislove. �


Corollary 3.2. Let (X,≤) be a directed poset with binary meets.Then (X,≤) is Hofmann-Mislove if and only if every maximalelement of an SCB prime set is prime.

Proof. By Theorem 3.2, it is sufficient to show that if (X,≤) hasfinite meets, then an SCB set, whose maximal elements are prime, isprime. Let K ⊆ X be an SCB set and suppose that every maximalelement ofK is prime. First we need to show thatK 6= X . Supposeconversely, that K = X . Then X is a directed set in the SCB setK, so X has an upper bound (and, of course, the greatest element)k ∈ K. Then k is a non-prime maximal element of K, which is acontradiction. Thus K 6= X . Now, let ↓ {a} ∩ ↓ {b} ⊆ ↓K = Kfor some a, b ∈ X . Then a ∧ b ∈ K, so there is a maximal elementm ∈ K such that a ∧ b ≤ m. By the assumption, m is prime, soa ≤ m or b ≤ m. Then a ∈ K or b ∈ K, which means that K isprime. �

The following corollary is the reformulated Hofmann-Mislove The-orem.

Corollary 3.3. Let (X,≤) be a non-empty distributive lattice. Then(X,≤) is Hofmann-Mislove.

Proof. Every non-empty poset with binary joins is directed, soCorollary 3.2 can be applied. Let K ⊆ X be an SCB prime set,m ∈ K a maximal element of K. Suppose that a ∧ b ≤ m forsome a, b ∈ X . Then (a ∨m) ∧ (b ∨m) = (a ∧ b) ∨m = m. Then↓ {a ∨m}∩↓ {b∨m} ⊆ K, but K is lower and prime, so a∨m ∈ Kor b ∨m ∈ K. But then a ∨m = m or b ∨m = m, i.e. a ≤ m orb ≤ m, since m is maximal. Hence, m is a prime element. �

On the other hand, the words ‘directed’ or ‘lattice’ cannot beomitted in Corollary 3.2 or Corollary 3.3, respectively, even if somekind of relaxed distributivity holds. The poset in the followingexample is close to a distributive lattice, but the top element ismissing. The distributive laws hold in this poset for those expres-sions which are correctly defined.

Example 3.1. Let (X,≤) be the poset with X = {1, 2, 3}, where1 ≤ 2, 1 ≤ 3 and 2, 3 are incomparable. The SCB prime sets are{1, 2}, {1, 3} and their maximal elements 2, 3 are prime.


On the other hand, X is a non-prime SCB set, whose maxi-mal elements are prime. By Theorem 3.2, (X,≤) is not Hofmann-Mislove. �

As we can also expect, a modular lattice need not be Hofmann-Mislove, which can be easily seen from the following example.

Figure 1.

Example 3.2. In the diamond lattice M5 depicted in Figure 1, theset {1, 2, 3} is prime and SCB. However, its maximal elements 2and 3 are not prime. Hence, the diamond lattice is not Hofmann-Mislove. �

On the other hand, there are Hofmann-Mislove lattices which arenot modular (and, of course, not distributive).

Example 3.3. The lattice (X,≤) on Figure 2 is not modular be-cause it has a pentagonal sublattice isomorphic to N5 with theunderlying set {−2, 0, 1, 2, 3}.

Figure 2.


It will be more illustrative if we show that (X,≤) is a Hofmann-Mislove lattice directly from the definition. The generalized spec-trum of this lattice is P = {0, 3, 6, 9, 12, . . .}, which is topologizedby the generalized hull-kernel topology

ωP = {∅, {0}, {0, 3}, {0, 3, 6}, {0, 3, 6, 9}, {0, 3, 6, 9, 12}, . . .}.

Then the up-filtered compact sets, which coincide with the compactsets since (X,≤) is a lattice, are precisely the finite sets. Hence,the compact saturated sets are exactly the elements of ωP . Thelattice (X,≤) is ∧-complete, so the filters in (X,≤) have the formF = ↑{f}, where f ∈ X . But not all the filters are SOB sets.The filters of the form ↑{3k} or ↑{3k+ 2} where k = 0, 1, 2, . . . arenot SOB sets, since the linearly ordered chain {1, 4, 7, . . .} has noupper bound in (X,≤), but it does not meet these filters (cf. Defini-tion 2.2). Therefore, the SOB filters are precisely the sets ↑{3k+1},where k ∈ {−1, 0, 1, . . .}. Let F = ↑{3k+1} be an SOB filter. ThenL(F ) =

⋂a∈F ψ(a) =

⋂a≥3k+1

(P r ↑{a}) = P r⋃

a≥3k+1↑{a} =

P r ↑{3k + 1} = P r F = {0, 3, 6, . . . , 3k} if k ∈ {0, 1, 2, . . .},or L(F ) = ∅ if k = −2 (cf. notation in Proposition 3.4). In anycase, L(F ) is compact saturated and {x | x ∈ X,L(F ) ⊆ ψ(x)} ={x | x ∈ X, ↑{x} ∩ P ⊆ ↑{3k + 1}} = ↑{3k + 1} = F . Con-versely, if L = {0, 3, 6, . . . , 3k} is a compact saturated set, the setF (L) = {x | x ∈ X,L ⊆ ψ(x)} = {x | x ∈ X, ↑{x}∩P ⊆ P rL} ={x | x ∈ X, ↑{x} ∩ P ⊆ ↑{3k + 1}} = ↑{3k + 1} is an SOB filter.Hence, by Definition 3.4, the lattice (X,≤) is Hofmann-Mislove. Ofcourse, alternatively one can prove that (X,≤) is Hofmann-Misloveby checking that the SCB prime sets are just the sets ↓{p}, wherep ∈ P , and then applying Corollary 3.2. �

It is well-known that the spectrum of a frame equipped with thehull-kernel topology is a sober topological space. One may ask whathappens with the sobriety of the generalized spectrum of a generalposet. We close the paper with an example, which shows thateven slightly more general Hofmann-Mislove posets than framesmay very naturally lead to non-sober topologies on their generalizedspectra.


Example 3.4. Let Y be an infinite set,X = {K | K ⊆ Y is finite}.For every a, b ∈ X we put a ≤ b if and only if a ⊇ b. Then (X,≤)has all finite meets including

∧∅ =

⋃∅ = ∅ ∈ X , which is the

top element of (X,≤). On the other hand,∨

∅ =⋂

∅ = Y is not afinite set, so (X,≤) has not the empty join. In particular, (X,≤) isa distributive lattice with all non-empty joins, it is a DCPO sincedirected sets are non-empty, but it is not a frame.

Let p = {y}, where y ∈ Y and suppose that a ∧ b ≤ p for somea, b ∈ X . Then a∪ b ⊇ p, which means that y ∈ a or y ∈ b. Hence,a ⊇ p or b ⊇ p, which gives a ≤ p or b ≤ p. Then p is a primeelement of X . Conversely, let p ∈ X be an element with |p| ≥ 2.Then there exist x, y ∈ p such that x 6= y. We put a = p r {x},b = p r {y}. We have a ∪ b = p, which implies a ∧ b ≤ p, but alsoa � p and b � p. It means that p is not prime. Since ∅ is not primeby the definition as the top element, the prime elements of (X,≤)are precisely the singletons. By Corollary 3.3, (X,≤) is a Hofmann-Mislove poset and its generalized spectrum is P = {{y} | y ∈ Y }.For every a ∈ X , P ∩ ↑{a} = {f | f ⊆ a, |f | = 1} = {{y} | y ∈ a}.Now we can see that the generalized hull-kernel topology on P isthe cofinite topology, which obviously is not sober. �

References

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2. Cameron, D. E., A survey of maximal topological spaces, Topology Proc. 2

(1977), 11-60.3. Gierz G., Hofmann K. H., Keimel K., Lawson J. D., Mislove M., Scott D. S.

A Compendium of Continuous Lattices, Springer-Verlag, Berlin, Heidelberg,New York, 1980, pp. 372.

4. Keimel K., Paseka J., A direct proof of the Hofmann-Mislove theorem, Proc.Amer. Math. Soc. 120 (1994), 301-303.

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8. Vickers S., Topology Via Logic, Cambridge University Press, Cambridge,1989, pp. 200

Department of Mathematics, Faculty of electrical engineering

and communication, University of Technology, Technicka 8, Brno,

616 69, Czech Republic

E-mail address: [email protected]

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