83

CONTRIBUTIONS TO GENERAL ALGEBRA 17Proceedings of the Vienna Conference 2005 (AAA70)

Verlag Johannes Heyn, Klagenfurt 2006

PRIME DECOMPOSITION ANDPSEUDOCOMPLEMENTATION

MARCEL ERNE

Abstract. In algebra, topology and order theory, decompositions intoprime elements or substructures play a crucial role. Often, the mathemat-ical objects in question carry an order structure such that each elementx has a ∨-pseudocomplement, that is, a least element whose supremumwith x gives the top element. In such posets P , one finds that the essen-tial primes, i.e. those join-prime elements whose ∨-pseudocomplement isnot the least element, are just the atoms of the skeleton P∗ (the Booleanposet of all pseudocomplements). A convenient necessary and sufficientcondition for the existence of a (unique) irredundant prime decompositionof the top element 1 is that P \{1} has a cofinal subset not containing anybinary forks (a specific kind of binary trees). In that case, the Dedekind-MacNeille completion of P∗ is isomorphic to the powerset of the irredun-dant prime decomposition. The exclusion of infinite upper antichains evenguarantees a least finite prime decomposition. Our results also providesome interesting consequences concerning the cellularity of ordered setsand topological spaces. As expected, relative pseudocomplements entailstill stronger decomposition properties. For example, the existence of atleast one essential prime for each non-zero element in a Brouwerian ∨-semilattice already guarantees unique irredundant prime decompositionsfor all elements.

0. Introduction

One of the most prominent links between order and topology was madepopular by Alexandroff in the thirties, when he pointed out the one-to-onecorrespondence between quasi-ordered sets, i.e. sets with a reflexive and tran-sitive relation, and Alexandroff spaces, i.e. topological spaces whose topology isclosed under arbitrary intersections (Alexandroff called them discrete spaces ;see [1]). Today, one formulates that correspondence between ordered andtopological structures by saying that the category of Alexandroff spaces (withcontinuous maps) is concretely isomorphic to the category of quasi-ordered

2000 Mathematics Subject Classification. Primary 06D15; secondary 06D20, 06F10.Key words and phrases. Antichain, cofinal, coframe, decomposition, essential, fork, irre-

dundant, prime, pseudocomplement, spoon.

84 M. ERNE

sets (with order-preserving = isotone maps). The isomorphism functor asso-ciates with any quasi-ordered set Q = (X,≤) the space AQ, whose open setsare the upper sets or up-sets

U = ↑U = {x ∈ X : u ≤ x for some u ∈ U},while the lattice of closed sets, also referred to as the Alexandroff completionof Q and denoted by AQ, consists of all lower sets or down-sets

D = ↓D = {x ∈ X : x ≤ a for some a ∈ D}.Frequently it is helpful to proceed along the following ‘recycling scheme’:

quasi-ordered sets

partially ordered sets

complete lattices

algebraic lattices

spatial coframes

algebraic coframes

closure systems

algebraic closure systems

topological closure systems

Alexandroff topologies

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An algebraic or compactly generated lattice is a complete lattice in which everyelement is a join of compact elements [2, 6], while a spatial coframe is a com-plete lattice in which every element is a join of ∨-prime elements (see, e.g.,[24] for the dual notion). It is well known that, up to isomorphism,

• the complete lattices are the closure systems,• the algebraic lattices are the algebraic closure systems,• the spatial coframes are the topological closure systems,• and the algebraic coframes are the Alexandroff topologies.

One of the earliest references to that theme is Buchi [4]. Recall that a frameor locale is a complete lattice enjoying the distributive law

x ∧∨

Y =∨{x ∧ y : y ∈ Y }

for arbitrary elements x and subsets Y , and a coframe satisfies the dual dis-tributive law. Algebraic spatial coframes are the same thing as superalge-braic lattices, characterized by the property that they have a join-dense setof completely join-prime (

∨-prime) elements, or dually, a meet-dense set of

completely meet-prime (∧

-prime) elements [12]. The Axiom of Choice (AC)allows to omit the word ‘spatial’ in the above characterization, on account of

PRIME DECOMPOSITION AND PSEUDOCOMPLEMENTATION 85

the Birkhoff-Frink Theorem about meet-decompositions in algebraic lattices[2]. Thus, assuming AC, the class of algebraic coframes is closed under dual-ization; in other words, an algebraic coframe is also a coalgebraic frame, andconversely.

The big jump from partially ordered sets to algebraic coframes may bedivided into several intermediate steps:

• ∨-pseudocomplemented posets• sectionally ∨-pseudocomplemented posets• relatively ∨-pseudocomplemented ∨-semilattices• dual Brouwerian lattices = dual Heyting algebras• coframes• spatial coframes• algebraic coframes

In the present study, we shall be concerned mainly with the most compre-hensive of these special classes of ordered sets, namely that of ∨-pseudocom-plemented posets (see below), because it includes all of the other importantstructures such as (dual) Heyting algebras, in particular Boolean algebras(with applications to mathematical logic, set theory and topology), and theircomplete version, the coframes. The class of spatial coframes in turn com-prises all closed set lattices of topological spaces, but also the dual lattices ofradical ideals in rings and other ideal structures, as well as down-set latticesof arbitrary ordered sets. Of course, there are also interesting classes of ∨-pseudocomplemented complete lattices that fail to be coframes, for example,the convex geometries.

By a ∨-pseudocomplemented poset we mean a poset that contains a topelement 1 and for each element x a ∨-pseudocomplement , that is, a least ele-ment y such that the only upper bound of x and y is 1, written x∨ y = 1 (see[22, 25, 33] for the dual notion). In a sectionally ∨-pseudocomplemented poset ,each principal ideal ↓ a = {x ∈ P : x ≤ a} is ∨-pseudocomplemented. Ex-amples of ∨-pseudocomplemented but not sectionally ∨-pseudocomplementedcomplete lattices are easily obtained by adding a new top element to anycomplete lattice that is not ∨-pseudocomplemented.

A ∨-semilattice S is Brouwerian or relatively pseudocomplemented if for allx, y ∈ S there is a relative pseudocomplement x\y such that

x ≤ y ∨ z ⇔ x\y ≤ z.

Obviously, such semilattices are sectionally ∨-pseudocomplemented, but notconversely: in fact, the finite Brouwerian semilattices are just the finite dis-tributive lattices, while many finite lattices like the pentagon are sectionally ∨-pseudocomplemented but not distributive. The simplest example of a Brouw-erian ∨-semilattice that fails to be a lattice (hence a dual Heyting algebra) is

86 M. ERNE

obtained by forming the ordinal sum of the chain ω of natural numbers anda three-element join-semilattice that is not a chain. Non-complete boundedchains are Brouwerian lattices but clearly not coframes. Non-atomic completeBoolean lattices like that of all regular open subsets of the reals are examplesof non-spatial (co)frames. Finally, most of the spatial coframes of closed setsoccurring in classical topology fail to be Alexandroff topologies, hence cannotbe algebraic.

In all, we see that we actually have a sequence of proper specializations,starting with ∨-pseudocomplemented posets, and ending with algebraic co-frames, concretely represented by Alexandroff topologies.

Our approach via joins is in accordance with the work by E.T. Schmidt [31,32] and is the adequate one for the purposes of a universal join decompositiontheory, opposite to the algebraic meet decomposition theory (see Crawley andDilworth [6]). Join decompositions are the more intuitive ones, because theelements or objects are ‘decomposed into smaller parts’ (note that also in[6], direct decompositions deal with joins rather than meets). Nevertheless, itshould be mentioned that the majority of authors prefer the dual notions like(∧-)pseudocomplements etc. (see e. g. Birkhoff [2], Frink [18], Gratzer [21],Katrinak [25, 26]).

The material of the present paper is organized as follows.Section 1 contains the crucial definition of κ-prime elements and a brief

review of recent results about prime decompositions in arbitrary posets (see[14] for more background).

In Section 2, we recall some of the basic definitions and results concerning∨-pseudocomplemented posets. In particular, we give a short proof for theknown fact that the skeleton (which consists of all ∨-pseudocomplements) hasa Boolean normal completion and is, therefore, a Boolean poset. Moreover,we demonstrate that if a ∨-pseudocomplemented poset P is meet-dense in acomplete lattice having the same top element as P , then that lattice is ∨-pseudocomplemented, too, and its skeleton is the normal completion of theskeleton of P (Theorem 2).

In Section 3 we show that for any maximal upper antichain (no two elementsof which have a common upper bound in the entire poset), the set of its ∨-pseudocomplements yields an irredundant decomposition of the top element.This has some interesting consequences for the cellularity of a poset or space.

In Section 4, we provide a whole series of equivalent characterizations foressential primes in a ∨-pseudocomplemented poset P by means of the skeletonP∗ (Theorem 4); here, an element of P is essential iff its ∨-pseudocomplementis not the least element.

Then, in Section 5, we turn to the decomposition theory for ∨-pseudocom-plemented posets. We show that the top element 1 of P has an irredundant

PRIME DECOMPOSITION AND PSEUDOCOMPLEMENTATION 87

prime decomposition iff the remainder P−1 has a cofinal subset Q of ‘spoonhan-dles’, i.e. elements q for which the principal dual ideal ↑q = {x ∈ Q : q ≤ x}is directed (Theorem 5). More precisely, any irredundant prime decomposi-tion arises as the set of all ∨-pseudocomplements of any upper antichain ofspoonhandles, and it consists of all atoms of the skeleton (Theorem 5). As aby-product, we find that the cellularity of P (the supremum of all cardinalitiesof upper antichains in P−1 ) is attained by any maximal upper antichain ofspoonhandles in (a cofinal subset of) P−1 and coincides with the cardinalityof the unique irredundant prime decomposition of the top element. Moreover,we show that 1 has a (least) finite prime decomposition iff P−1 contains noinfinite upper antichains (Theorem 2). This result includes a famous theoremdue to Erdos and Tarski [8] about the cellularity. Analogous results hold forκ-primes (Theorem 7).

Finally, in Section 6, we have a closer look at Brouwerian semilattices. Itfollows from the above results (by relativization to principal ideals) that aBrouwerian ∨-semilattice is free over the poset of its primes iff no truncatedprincipal ideal contains an infinite upper antichain, which is certainly fulfilledif the whole semilattice does not contain any infinite antichain. In dual Heytingalgebras, it suffices to require a meet-dense subset without infinite antichainsin order to assure least finite prime decompositions for all elements. Thisgeneralizes the known decomposition of ordered sets with no infinite antichainsinto a finite number of ideals (see Bonnet [3], Diestel [7]).

1. Prime Decompositions and Distributivity in Posets

In order to have the necessary tools at hand, we start with a few generaldefinitions and remarks about prime elements and decompositions in arbitraryposets. For a more systematic treatment of that topic, we refer to [14].

Let κ be any cardinal number or the symbol ∞, and write A ⊂κ B if Ais a subset of B with fewer than κ elements (where A ⊂∞ B simply meansthat A is a subset of B, i.e. A ⊆ B). The κ-directed subsets D of a posetor quasi-ordered set P are characterized by the property that each subset ofD with fewer than κ elements has an upper bound in D. For 2 < κ ≤ ω,it is clear that ‘κ-directed’ means ‘directed’, and an ∞-directed set is onewith a greatest element. An element p ∈ P is said to be κ-prime if the set{x ∈ P : p 6≤ x} is κ-directed. In a ∨-semilattice, the κ-primes with 2 < κ ≤ ωare the join-primes (∨-primes), and in a complete lattice, the ∞-primes arethe completely join-primes (

∨-primes) in the usual sense.

A join-decomposition of an element x of a poset P is a subset with join x,and a prime decomposition is a join-decomposition that consists of ω-primes.If A is a set and a is an element of A, it will be convenient to write A− a forA \{a}. A subset A of a poset P is called irredundant if for each a ∈ A, thereis an upper bound b of A−a with a 6≤ b ; the latter simply means a 6≤

∨(A−a)

88 M. ERNE

if that join exists (cf. [6, Ch.7]). Thus, a join-decomposition is irredundant iffit is minimal with respect to inclusion. Every irredundant set is an antichain(a set of pairwise incomparable elements), but not conversely. Stronger thanirredundance is the following property: a subset A of P is independent if foreach a in A, there is an upper bound b of A− a such that a and b are disjoint,and a is not the least element of P . In complete lattices, independence meansa∧

∨(A−a) = 0 6= a for all a ∈ A. An independent join-decomposition is also

referred to as a direct decomposition (cf. [6, Ch.8]). Every independent set isirredundant and has pairwise disjoint elements (where disjointness in posetsmeans that there is no common lower bound, except a possible least element).Conversely, in a frame, a set of pairwise disjoint non-zero elements is alreadyindependent. (Caution: In the theory of Boolean algebras, ‘irredundance’,‘independence’ and ‘antichains’ sometimes have a different meaning; see e.g.[27].)

We call an element p ∈ P ∨-essential for an element x ∈ P if there existsa q < x with x = p ∨ q. Slightly weaker is the following property: an elementp is essential for x if there is a q with x 6≤ q but ↑p ∩ ↑q ⊆ ↑x (which meansx ≤ p∨q if that join exists). In case x is the greatest element, both conditionsare equivalent, and we simply speak of an essential element (see [14]; cf. [7]for the more specific notion of essential ideals, and [30] for the dual notion ofessential primes in frames). Any irredundant join-decomposition of x consistsof essential elements for x. Essential primes are maximal among all primes,but not conversely. However, if an element x has a finite prime decompositionthen the essential primes for x are precisely the maximal primes below x. Thefollowing strong uniqueness theorem for irredundant prime decompositions hasbeen established in [14]:

Theorem 1. Let R be a subset of a poset P and call R-prime those membersof R which are ω-prime in P . Then the following are equivalent for any subsetA of P and an element x having at least one join-decomposition into R-primes:

(a) A is the set of all essential R-primes for x and has join x.(b) A is the least join-decomposition of x into maximal R-primes in ↓x.(c) A is an irredundant join-decomposition of x into R-primes.(d) A consists of maximal R-primes and is maximal irredundant in ↓x.

If x has a finite join-decomposition into R-primes then these conditions arealso equivalent to each the following statements:

(aω) A is the set of all maximal R-primes in ↓x and has join x.(bω) A is the least finite join-decomposition of x into R-primes.(cω) A is a finite antichain of R-primes with join x.(dω) A is a maximal irredundant set of maximal R-primes in ↓x.

PRIME DECOMPOSITION AND PSEUDOCOMPLEMENTATION 89

Note the subtle difference between (d) and (dω): the former condition in-cludes that A is maximal among all irredundant sets consisting of elementsbelow x, while in the latter, it is only required that A be maximal among theirredundant sets of maximal R-primes below x.

Analogous results hold for κ-prime decompositions (see [14] for details).In the late seventies, we have initiated a theory of distributive laws for or-

dered sets (see [9, 10, 11, 12]); later on, the theory was extended from orderedsets to closure spaces, contexts and their concept lattices (see [13, 16]). In-dependently, similar results about modular, distributive and Boolean orderedsets were obtained by Chajda, Larmerova, Rachunek [5, 28] and Niederle [29].

Let us briefly recall some of the main ideas and concepts in that area.As is well known, any poset P is join- and meet-densely embedded in anup to isomorphism unique complete lattice, its completion by cuts, Dedekind-MacNeille completion or normal completion NP (see, for example, [2] and[10, 11]). Concretely, NP may be regarded as the closure system of all cuts.The corresponding closure operator ∆ sends each subset A to the intersectionof all principal ideals containing A. Denoting by A↓ and A↑ the set of all lowerand upper bounds of A, respectively, we have ∆A = A↑

↓. The embedding of Pin NP associates with each x ∈ P the principal ideal ↓x = {x}↓. However, itis more convenient to consider P as a subposet of NP . The lattice extension(in [29] : characteristic lattice) of P is then the lattice LP generated by Pin NP . The following basic result characterizes so-called (weakly) distributiveordered sets by various alternative conditions and shows that distributivity isa self-dual property, also in the general setting of posets.

Proposition 1. For any poset P , the following are equivalent:(a) ↓x ∩∆{y, z} = ∆(↓x ∩ ↓{y, z}) for all x, y, z ∈ P .(b) ↓x ∩∆Y = ∆(↓x ∩ ↓Y ) for all x ∈ P and all finite Y ⊆ P .(c) For finite Y ⊆ P and x ∈ ∆Y , there is a Z ⊆ ↓Y with x =

∨Z.

(d) ∆ preserves finite intersections of finitely generated down-sets.(e) The lattice extension LP is distributive.(f ) In NP , the identity x∧ (y ∨ z) = (x∧ y)∨ (x∧ z) holds for x, y, z ∈ P .(g) For all x, y, z ∈ P , ↓x ∩ ↓y ⊆ ↓z and ↑x ∩ ↑z ⊆ ↑y imply y ≤ z.

Stronger are the following mutually equivalent conditions:(h) P is strongly distributive, i.e. ↓x ∩ ∆Y = ∆(↓x∩ ↓Y ) for all Y ⊆P .( i ) For all Y ⊆ P and x ∈ ∆Y , there is a Z ⊆↓Y with x =

∨Z.

( j ) ∆ preserves finite intersections of down-sets.(k) The normal completion NP is a frame.

See [10] and [13] for these and related facts concerning posets with distribu-tive completions by cuts or ideals, respectively.

90 M. ERNE

Note that every κ-prime element p is κ-irreducible (i.e. A⊂κ P and p =∨

Aimply p ∈ A), and that the converse holds in strongly distributive posets (infact, for κ ≤ ω , weak distributivity suffices).

2. Pseudocomplemented and Boolean Posets

Two elements a and a′ of P are called disjoint (respectively, cojoint) if↓a ∩ ↓a′ = P↓ (respectively, ↑a ∩ ↑a′ = P ↑). If a and a′ are both disjoint andcojoint then they are complements of each other. A poset P is complemented ifeach of its elements has at least one complement. Distributive complementedposets are called Boolean (cf. [29]). Furthermore, a poset P is (∧-)pseudocom-plemented (cf. [22, 25, 26, 29, 33]) if for each a ∈ P there exists a greatestelement a∗ disjoint from a, called the (∧-)pseudocomplement of a . The ∨-pseudocomplement a∗ and ∨-pseudocomplemented posets are defined dually.Usually one postulates a least element 0 and a greatest element 1 for (pseudo-)complemented posets; in that case, a and a′ are disjoint (cojoint) iff a∧a′ = 0(a ∨ a′ = 1). It is convenient to write A∗ for {a∗ : a ∈ A} and A∗ for{a∗ : a ∈ A} (A ⊆ P ). The poset P ∗ (respectively, P∗) is called the (∨-)Booleanization or the (∨-)skeleton of P (see [2, 18, 25, 26, 29]).

The famous Glivenko-Frink Theorem [18], stating that the skeleton of apseudocomplemented semilattice S is always Boolean, extends to the posetsetting, as demonstrated by Halas [22, 23] and Niederle [29]. Almost 30 yearsearlier, Katrinak [25] had already shown that the skeleton of a pseudocomple-mented poset is a Boolean lattice iff it is a ∨- or ∧-semilattice. We add herea short alternative proof of a slightly stronger result.

Proposition 2. If P is a pseudocomplemented poset then the skeleton P ∗ isstrongly distributive and uniquely complemented, in particular Boolean.Hence, the following assertions are equivalent:

(a) P is Boolean.(b) P is uniquely complemented.(c) P coincides with its (∨-)skeleton.

Proof. The operators ∆ and ↓ refer to P ∗. Suppose x∈∆Y for some Y ⊆P ∗.Given ↓ x ∩ ↓ Y ⊆↓ z∗, we have to show that x ≤ z∗ (then Z = ↓ x∩ ↓ Ysatisfies x =

∨P ∗Z). If b is a lower bound of {x, z} in P ∗ and y is an element

of Y , then b and y are disjoint, because ↓b∩ ↓y ⊆↓z ∩ ↓x ∩ ↓Y ⊆ ↓z ∩ ↓z∗.Hence, we have y ≤ b∗ for all y ∈ Y , and the hypothesis x ∈ ∆Y yieldsb ≤ x ≤ b∗ and so b = 0. Thus, x and z are disjoint, and in fact x ≤ z∗.

For x ∈ P ∗, the pseudocomplement x∗ is easily seen to be the uniquecomplement in P ∗ (see [22] and the remarks below). Since the map x 7→ x∗ isa dual automorphism of P ∗, it is clear that P ∗ must be a lattice whenever itis a ∨- or ∧-semilattice.

PRIME DECOMPOSITION AND PSEUDOCOMPLEMENTATION 91

For the equivalence of (a), (b) and (c), see also Niederle [29]. 2

In view of the intended applications to join-decompositions, from now on weadopt the general assumption that

P is a ∨-pseudocomplemented poset and P∗ its ∨-skeleton.Let us point out some properties of such posets. P has a top element 1 iff ithas a bottom element 0, in which case 0 = 1∗, and 1 is the only element awith a∗ = 0. We shall make frequent use of the basic equivalences

a ∨ b = 1 in P ⇔ a∗ ≤ b ⇔ b∗ ≤ a ⇔ b∗ ≤ a∗∗ ⇔ a∗ ∧ b∗ = 0 in P∗.

From these equivalences it follows at once that for each a ∈ P∗ the ∨-pseudo-complement a∗ is the unique complement in P∗. The map a 7→ a∗∗ is a kerneloperator (i.e. a∗∗ ≤ b ⇔ a∗∗ ≤ b∗∗), and consequently, in the range P∗ = P∗∗,joins coincide with those in P .

It is known that the normal completion of a Boolean lattice or poset is againBoolean (see [2, Ch.V.11], [23] and [29]), and that the normal completion ofa pseudocomplemented poset is again pseudocomplemented (see [13], [16] and[23]). Stronger is the following theorem, parts of which had been obtained, ina dual form using the language of standard completions, in [16]:

Theorem 2. If a ∨-pseudocomplemented poset P is meet-dense in a completelattice L whose top element belongs to P , then L is ∨-pseudocomplemented,too, and ∨-pseudocomplements in P coincide with those in L. Furthermore,

L∗ ' N (P∗) ' (NP )∗is a complete Boolean algebra. Hence, if P is Boolean then so is NP .

Proof. Meet-density of P entails that joins in P coincide with those in L, andthat for c ∈ L, the element c∗ =

∨L{a∗ : a ∈ P, c ≤ a} is the ∨-pseudo-

complement in L; indeed, for any d ∈ L,c∗ ≤ d ⇔ ∀a ∈ P (c ≤ a ⇒ a∗ ≤ d) ⇔∀a, b ∈ P (c ≤ a, d ≤ b ⇒ a∗ ≤ b) ⇔∀a, b ∈ P (c ≤ a, d ≤ b ⇒ a ∨ b = 1) ⇔ c ∨ d = 1.

In order to prove meet-density of P∗ in L∗, choose a c ∈ L and a set A ⊆ Pwith c∗ =

∧L A. Then A∗∗ ⊆ P∗ and c∗ =

∧L∗

A∗∗. But P∗ is also join-densein L∗, since c =

∧B with B ⊆ P implies B∗ ⊆ P∗ and c∗ =

∨L∗

B∗: ford ∈ L∗, we have c∗ ≤ d ⇔ d∗ ≤ c ⇔ ∀b ∈ B (d∗ ≤ b) ⇔ ∀b ∈ B (b∗ ≤ d).

In all, we see that P∗ is both join- and meet-dense in L∗, so that L∗ is(isomorphic to) the normal completion of P∗. The isomorphism (NP )∗ 'N (P∗) is now immediate, by considering the special case L = NP . By the dualof Proposition 2, (NP )∗ is Boolean, because NP is ∨-pseudocomplemented bythe first part of the proof. Finally, if P is Boolean then P = P∗, and thereforeNP = N (P∗) = (NP )∗ is Boolean, too. 2

92 M. ERNE

The above considerations also show that if P is κ-join-complete then so isP∗, and in particular, that P∗ is a Boolean lattice if P is a ∨-semilattice.

Observe that a finite poset with Boolean normal completion need not be(∨-)pseudocomplemented. Counterexamples were given in [16] and [23].

3. Antichains and Irredundant Decompositions

In Boolean lattices, the ∨-primes are just the atoms (that is, the mini-mal non-zero elements). Surprisingly, much of the decomposition theory forBoolean algebras and for topological spaces extends to the considerably moregeneral setting of pseudocomplemented semilattices or posets (no distributivelaws assumed).

The following definition will be crucial for our investigation of join-decom-positions: an upper (lower) antichain in a poset is a subset of elements notwo of which have a common upper (lower) bound in the entire poset. Upperantichains are also termed strong antichains. Clearly, such sets are antichains,and every set of maximal elements is an upper antichain, but neither of theseimplications can be inverted. The system of all upper (lower) antichains is offinite character, so by Zorn’s or Tukey’s Lemma, every upper (lower) antichainis contained in a maximal one. Henceforth, suppose that

P is a ∨-pseudocomplemented poset with greatest element 1, andQ is a cofinal subset of P−1, that is, ↓Q = P−1.

Consequently, a subset A of P has an upper bound in Q iff 1 is not the joinof A in P , and Q∗ is coinitial (i.e. dually cofinal) in P∗−0. Note also that thesets of pairwise disjoint (cojoint) elements in P are precisely the lower (upper)antichains in P− 0 (P−1). Let us call a subset A of P ∗-faithful if the mapa 7→ a∗ from A onto A∗ is one-to-one. Every upper antichain A in P −1 is∗-faithful, because for distinct a, b ∈ A, we have a∗ ≤ b but b∗ 6≤ b. Moreover,every subset of a Boolean lattice is ∗-faithful (since a 7→ a∗ is then bijective onthe whole lattice). We are now ready for a useful generalization of the knownresult that in a Boolean lattice, the maximal pairwise disjoint families are thedirect decompositions of the top element (cf. [27, Ch.1,3.6]).

Theorem 3. Consider the following conditions on a set A ⊆ Q:(a) A is a maximal upper antichain in Q.(b) A∗ is a maximal lower antichain in Q∗.(c) A∗ is a lower antichain in Q∗ with

∨A∗ = 1 (in P or in P∗).

(d) A∗ is a direct decomposition of 1 (‘partition of unity’) in P∗.(e) A∗ is an irredundant join-decomposition of 1 in P .

The implications (a) ⇒ (b) ⇔ (c) ⇔ (d) ⇒ (e) are always true.If A is ∗-faithful, the first four conditions (a) – (d) are equivalent.

PRIME DECOMPOSITION AND PSEUDOCOMPLEMENTATION 93

Proof. (a) ⇒ (b). Use the equivalence a ∨ b = 1 in P ⇔ a∗ ∧ b∗ = 0 in P∗.(b) ⇒ (d). If A∗ had an upper bound b < 1, we could assume b ∈ Q. Then

a∗ ≤ b , i.e. a∗ ∧ b∗ = 0 in P∗ for all a ∈ A, whence b∗ ∈ A∗ by maximalityof A∗. But then b∗ ≤ b = b ∨ b∗ = 1, a contradiction. The decomposition1 =

∨A∗ in P∗ is direct, because for each a ∈ A the element a∗∗ satisfies

a∗∧a∗∗ = 0 in P∗ and is an upper bound of A∗−a∗; indeed, b ∈ A and b∗ 6= a∗imply a∗ ∧ b∗ = 0 in P∗, hence b∗ ≤ a∗∗.

(d) ⇒ (c). Clear, since 0 6∈ Q∗ and joins in P∗ agree with those in P .(c) ⇒ (b). If some b ∈ Q would satisfy a∗∧ b∗ = 0 in P∗ for all a ∈ A, then

a∗ ≤ b for all a ∈ A and 1 =∨

A∗ ≤ b, which is excluded by Q ⊆ P−1.(d) ⇒ (e). If b < 1 would be an upper bound of A∗ in P then b∗∗ would be

an upper bound of A∗ in P∗−1. Clearly, A∗ is an irredundant lower antichainin Q∗, being a direct decomposition.

Under the hypothesis that A is ∗-faithful, we derive (a) from (c).For distinct a, b ∈ A, we have a∗ 6= b∗ and therefore a∗ ∧ b∗ = 0 in P∗, hencea∨ b = 1. Thus A is an upper antichain. Concerning maximality, observe thatno b ∈ Q can satisfy a ∨ b = 1 for all a ∈ A, because the latter would entail1 =

∨A∗ ≤ b. 2

It is plain that (b) does not imply (a) in general (consider a chain with morethan two elements). Furthermore, even in finite Boolean lattices, condition (e)is strictly weaker than the other ones: for any set A of two atoms, the set A∗is an irredundant join-decomposition of the top element (into coatoms), butnot a direct decomposition if there are more than two atoms.

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Generalizing the notion of cellularity for topological spaces and Booleanalgebras (see [27, Ch.1.3]), one defines the (lower) cellularity of a poset P tobe the supremum of all cardinalities of subsets of pairwise disjoint elements.The upper cellularity of a poset is the lower cellularity of the dual poset (cf.[27, Ch.2.4], where a slightly different definition of cellularity is given). As animmediate consequence of Theorem 3, we note:

Proposition 3. The upper cellularity of P (and of Q ∪ {1}) coincides withthe upper and the lower cellularity of the self-dual skeleton P∗; hence, it is thesupremum of all cardinalities of direct decompositions for 1 in P∗.

The dual of that result is an abstract order-theoretical generalization ofthe fact that the cellularity of a topology is the same as the cellularity of thecomplete Boolean algebra of its regular open sets.

94 M. ERNE

An analogous equality holds for the lower (upper) saturation, the leastcardinal number greater than all cardinalities of pairwise disjoint (cojoint)subsets (see Erdos and Tarski [8] and [27, Ch.1.3]).

4. Spoons, Forks and Primes

Well-known characterizations of atoms in Boolean algebras extend to thesituation of ∨-pseudocomplemented posets. Moreover, there is an interestingconnection between atoms, primes and so-called ‘spoonhandles’, i.e. elementsq ∈ Q for which the principal dual ideal ↑Q q = {x ∈ Q : q ≤ x} is directed (a‘spoon’). We prove a more general result on κ-primes and κ-spoonhandles , i.e.elements generating a κ-directed principal dual ideal. By cofinality of Q in P ,an element q is a κ-spoonhandle in Q iff 1 is κ-prime in ↑P q = {x ∈ P : q ≤ x}(for κ > ω, this equivalence requires choice).

Proposition 4. Suppose p and r are elements of P with r < 1 = p ∨ r.

(1) If p is κ-prime then ↑Q r = {x ∈ Q : r ≤ x} is κ-directed.(2) Conversely, if ↑Q r is κ-directed then r∗ is κ-prime.(3) An element q ∈ Q is a κ-spoonhandle in Q iff q∗ is κ-prime in P (and

also in P∗).

Proof. (1) Suppose A ⊂κ ↑Q r. Then each a ∈ A fulfils r ≤ a 6= 1, hence p 6≤ a(as p∨r = 1). By κ-primeness of p, we find an upper bound b of A with p 6≤ b,in particular b 6= 1. This b is dominated by some element of Q, which is thenan upper bound of A in ↑Q r, provided A is nonempty. But if A = ∅ then anyelement of ↑Q r (which exists by cofinality of Q in P−1) is trivially an upperbound of A in ↑Q r.

(2) Assume ↑Q r is κ-directed, and put p = r∗. If A ⊂κ P and p 6≤ a forall a ∈ A then there is a set B = {ba : a ∈ A} ⊂κ Q such that each ba is anupper bound of {a, r} (use cofinality of Q in P−1 and the Axiom of Choice).It follows that B ⊂κ↑Q r , whence B has an upper bound b in Q, which is thenalso an upper bound of A. It cannot happen that p ≤ b , because otherwise1 = p ∨ r ≤ b . This shows that p is κ-prime in P .

(3) is a consequence of (1) and (2). If A ⊂κ P∗ then A ⊆↓ b impliesA = A∗∗ ⊆↓ b∗∗. Since q∗ 6≤ b is equivalent to q∗ 6≤ b∗∗, we see that wheneverq∗ is κ-pime in P , it is also κ-prime in P∗. 2

Note that an element p ∈ P is essential (i.e. p ∨ r = 1 for some r < 1)iff p∗ is not the top element. Clearly, in a Boolean algebra, all ∨-primes areatoms and essential. But observe that, in contrast to the case of Booleanalgebras, a (finitely) ∨-irreducible element of a Boolean poset P = P∗ neednot be (completely)

∨-irreducible (meaning that it need not belong to every

subset whose join it is). This discrepancy between the poset and the latticecase is witnessed by

PRIME DECOMPOSITION AND PSEUDOCOMPLEMENTATION 95

Example 1. Take top and bottom, atoms and coatoms of an infinite powersetlattice. This gives a Boolean poset whose coatoms are ∨-irreducible but not∨

-irreducible.

Now to the characterizations of atoms in the ∨-skeleton P∗.

Theorem 4. For p ∈ P and 2 < κ ≤ ∞, the following are equivalent:(a) p is an atom of P∗.(b) p is κ-prime in P∗.(c) p is

∨-irreducible in P∗ (i.e. p =

∨A in P∗ implies p ∈ A).

(d) p is ∨-prime in P∗ (i.e. if A ⊂ω P∗ has a join above p then p ∈↓A) .(e) p is ω-prime in P and an element of P∗.(f ) p is an essential ω-prime of P .(g) p is the ∨-pseudocomplement of a spoonhandle in Q.(h) p = q∗ for some q ∈ Q such that 1 is ω-prime in ↑P q.

Proof. (a) ⇒ (b). The equivalences p 6≤ x ⇔ p ∧ x = 0 in P∗ ⇔ x ≤ p∗ forx∈P∗ show that P∗\ ↑p is the principal ideal ↓p∗ = {x ∈ P∗ : x ≤ p∗}.

The implications (b) ⇒ (c) and (b) ⇒ (d) are obvious.(c) ⇒ (a). If p is

∨-irreducible in P∗ then also in the normal completion

N (P∗), since P∗ is join-dense in N (P∗). Hence, p is an atom of the Booleanalgebra N (P∗) and therefore also an atom of P∗.

(d) ⇒ (a). The assumption a∗ < p together with p ≤ 1 = a∗ ∨ a∗∗ leads toa∗ < p ≤ a∗∗, hence to a∗ = a∗ ∧ a∗∗ = 0 in P∗. Thus, p is an atom of P∗.

(a) ⇒ (e). It suffices to show that p = q∗ is 3-prime in P if it is an atom inP∗. Assume p 6≤ ai (i = 1, 2). Then there are bi < 1 with q ≤ bi and ai ≤ bi.It follows that 0 < bi∗ ≤ p and consequently p = b1∗ = b2∗. The hypothesisb1 < 1 entails a2∗ 6≤ b1 (otherwise 1 = q ∨ p = q ∨ b2∗ ≤ q ∨ a2∗ ≤ b1). Hence,there is a c < 1 such that a2 ≤ c and a1 ≤ b1 ≤ c. But q ≤ b1 ≤ c < 1 togetherwith p ∨ q = 1 excludes p ≤ c. Thus p is 3-prime and so ω-prime.

(e) ⇒ (f). p∗=1 would entail p=p∗∗ = 0, contradicting ω-primeness.(f) ⇒ (g). If p is an essential ω-prime in P then p∗ is a spoonhandle in

P−1, by Proposition 4 (1), applied to κ = ω, p∗ 6= 1 in place of r, and P−1instead of Q. Since Q is cofinal in P−1, we have p∗ ≤ q for some q ∈ Q, inparticular p 6≤ q (otherwise 1 = p∨ p∗ ≤ q). It follows that q is a spoonhandlein Q with q∗ = p: indeed, p∗ ≤ q yields q∗ ≤ p, and p 6≤ q together withp ≤ 1 = q ∨ q∗ entails p ≤ q∗, because p is ∨-prime.

From Proposition 4 (3), applied to κ = ω, we infer the implications (g)⇒ (e)and (e) ⇒ (b) with κ = ω. Finally, the equivalence of (g) and (h) is clear bycofinality of Q in P−1 and the definition of spoonhandles. 2

96 M. ERNE

By a fork in a poset, we mean a non-linearly ordered subset Y having aleast element but no incomparable elements with a common upper boundin the entire poset. (This notion has to be distinguished from that of a tree,where upper bounds are excluded only in the tree itself, and linearity is notforbidden).

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A binary fork is a countable fork Y = {yn : n ∈ N = ω \{0}} such thatym < yn iff 2km ≤ n < 2k(m + 1) for some k ∈ N.

Of course, any such binary fork contains infinite chains, but also infinite upperantichains, for example {y2k+1 : k ∈ N} or {y2k−2 : k ∈ N−1}.

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Lemma 1. For an arbitrary poset, the following are equivalent:(a) There exists a cofinal subset consisting of spoonhandles.(b) There exists a cofinal subset not containing any fork.(c) There exists a cofinal subset not containing any binary fork.

Each of these conditions is fulfilled if(d) there exists a cofinal subset without infinite upper antichains.

Proof. The implications (a) ⇒ (b) ⇒ (c) and (d) ⇒ (c) are obvious. Theimplication (c) ⇒ (a) is obtained by contraposition. If some ↑ q contains nospoonhandle then in every cofinal subset C, one may construct, using depen-dent choices, a binary fork: choose y1 ∈↑ q ∩ C; then ↑ y1 is not directed;choose y2 and y3 in ↑y1∩C with no common upper bound, and so on (cf. [7]).

2

We shall call a poset with the above equivalent properties spoonful.

PRIME DECOMPOSITION AND PSEUDOCOMPLEMENTATION 97

5. Irredundant Prime Decompositions

Now, we are prepared for the main results in this paper. They do not onlyinclude necessary and sufficient conditions for the existence of irredundantprime decompositions in lower pseudocomplemented posets, but also providethe ‘essential’ (and necessary) tools in order to find such decompositions.

Proposition 5. Let P be a ∨-pseudocomplemented poset with top element 1,and assume that P−1 has a cofinal subset Q of spoonhandles. Furthermore,let A be a maximal upper antichain A in Q. Then:

(1) The set A∗ is the least, hence unique irredundant prime decompositionof 1 in P and also the direct decomposition into atoms of the skeleton P∗.

(2) The upper cellularity of P is attained by A and is therefore equal to thecardinality of the least prime decomposition of 1.

(3) The normal completion N (P∗)' (NP )∗ is a complete atomic Booleanalgebra, isomorphic to the powersets of A and of A∗.

Proof. (1) By Theorem 4, A∗ consists of ω-primes in P , and these are atomsof P∗. By Theorem 3, A∗ is an irredundant join-decomposition of 1 in P anda direct decomposition in P∗. By Theorem 1, such decompositions are unique.

(2) If A′ is any upper antichain in P−1 then, by cofinality of Q, there isa (maximal) upper antichain A in Q with A′ ⊆↓A, and it follows that eacha′ ∈ A′ is dominated by a unique a ∈ A. This shows that the cardinality ofA′ is less than or equal to that of A, which in turn coincides with that of A∗(because A is ∗-faithful).

(3) This part is clear by (1) and the obvious fact that the atoms of P∗ areprecisely those of N (P∗). 2

From the previous facts, we now deduce a collection of necessary and sufficientcriteria for the existence of irredundant prime decompositions.

Theorem 5. For a ∨-pseudocomplemented poset P with top element 1, thefollowing are equivalent:

(a) P−1 is spoonful.(b) 1 has a (unique) irredundant prime decomposition in P .(c) 1 has a (unique) irredundant prime decomposition in P∗.(d) P∗ is atomistic (each element is a join of atoms).(e) P∗ is atomic (the set of atoms is coinitial in P∗−0).(f ) P∗ is coatomic (the set of coatoms is cofinal in P∗−1).(g) N (P∗) is a complete atomic Boolean algebra.

Sufficient but not necessary for these properties are the following conditions:(h) P is coatomic, i.e. P−1 has a cofinal antichain.( i ) P−1 has a cofinal subset with no infinite (ascending) chains.

98 M. ERNE

( j ) P−1 has a cofinal subset with no infinite antichains.(k) P−1 contains no infinite upper antichains.( l ) P−1 contains no binary fork.

Proof. For (a) ⇒ (b), apply Proposition 5 (1).(b) ⇒ (c). Use the fact that an irredundant prime decomposition is unique

and consists of essential ω-primes (Theorem 1), and that these are ω-primeatoms of the ∨-skeleton P∗ (Theorem 4).

(a) ⇒ (d). Assume x∗ 6≤ y∗ in P∗. Then there is a b < 1 with x ≤ band y∗ ≤ b, and by (a), we may assume that b is a spoonhandle in P −1.Hence, by Theorem 4, b∗ is an atom of P∗ with b∗ ≤ x∗ but b∗ 6≤ y∗ (otherwiseb = b ∨ b∗ = 1). Thus, x∗ is a join of atoms in P∗.

(c) ⇒ (e). If D is the irredundant prime decomposition of 1 in P∗ then,again by Theorems 1 and 4, D consists of atoms in P∗. Now, 0 < a in P∗entails a∗ < 1, hence p 6≤ a∗ for some p ∈ D, and then p ≤ a (otherwisea ∧ p = a∗∗ ∧ p∗∗ = 0 in P∗ and p = p∗∗ ≤ a∗).

(e) ⇒ (a). For b ∈ P−1, we have b∗ ∈ P∗−0, so there is an atom a ∈ P∗ witha ≤ b∗; by Theorem 4 and Proposition 4, it follows that a∗ is a spoonhandlein P−1. It cannot happen that a∗ ∨ b = 1 (otherwise a ≤ b∗ ≤ a∗ and thena∗ = a ∨ a∗ = 1, a = a∗∗ = 0). Hence, there is an upper bound c < 1 of a∗and b, and as a∗ is a spoonhandle in P−1, so is c.In all, we have established the implication circles

(a)⇒ (b)⇒ (c)⇒ (e)⇒ (a) and (a)⇒ (d)⇒ (e)⇒ (a).For (e)⇔ (f) use the dual automorphism a 7→ a∗ of P∗, and for (e)⇔ (g)

apply Theorem 2 about the normal completion of P∗.The implications (h)⇔ (i)⇒ (a) and (j)⇒ (k) ⇒ ( l ) ⇒ (a) are clear. 2

Replacing P with P∗ = P∗∗, we see that the conditions in Theorem 5 are alsoequivalent to the postulate that P∗−1 be spoonful.Notice that the exclusion of binary forks is a much stronger property than theexistence of a cofinal subset of P−1 containing no binary forks.

Example 2. The tree of all finite or infinite binary words (0-1-sequences),ordered by the prefix relation, has a cofinal upper antichain (viz. the set ofall infinite words), which clearly does not contain any binary fork. Adding atop element to the tree, one obtains a coatomic and ∨-pseudocomplementedcomplete lattice; for any word with first letter a, the singleton word 1 − a isthe ∨-pseudocomplement. In that lattice, conditions (h) and (i) are fulfilled,whereas (j), (k) and (l) are violated. On the other hand, the chain ω + 1satisfies the conditions (j), (k) and (l) but neither (h) nor (i).

Invoking Theorem 3 once more, we arrive at

PRIME DECOMPOSITION AND PSEUDOCOMPLEMENTATION 99

Corollary 1. If the equivalent conditions in Theorem 5 are fulfilled then theupper cellularity of P is, at the same time,

the upper and lower cellularity of P∗,the cardinality of the set of all essential primes in P ,the cardinality of the set of all (co)atoms in P∗,the cardinality of the irredundant prime decomposition of 1 in P ,the cardinality of any maximal upper antichain of spoonhandles in P−1.

Concerning finite decompositions, the previous considerations yield:

Corollary 2. For a ∨-pseudocomplemented poset P , the following are equiv-alent:

(a) P has finite upper cellularity.(b) P−1 contains no infinite upper antichains.(c) 1 has a finite prime decomposition.(d) 1 has a least, hence unique irredundant finite prime decomposition.(e) The skeleton P∗ is finite.(f ) The normal completion N (P∗) is a finite Boolean algebra.

If these conditions are fulfilled then the upper cellularity of P equals the numberof maximal (=essential) primes.

In (a) and (b), P−1 may be replaced with a cofinal subset Q (because the upperantichains in Q are also upper antichains in P−1 and any upper antichain inP−1 is dominated by an upper antichain in Q).

An old theorem due to Erdos and Tarski [8] states that the lower/uppercellularity of a poset Q is finite if Q does not contain any infinite subset ofpairwise disjoint/cojoint elements (in other words, that the saturation nevercan be ω). This result is an immediate consequence of Theorem 2, by takingfor P the down-set coframe of a poset Q and observing that the complementsof principal filters form a cofinal subset of AQ \{Q} that is isomorphic to Q.See also Proposition 3.

For semilattices, some of the previous results may be improved:

Corollary 3. The skeleton S∗ of a ∨-pseudocomplemented ∨-semilattice S isan atomic Boolean algebra iff S−1 is spoonful. Moreover, S∗ is a finite Booleanalgebra iff S−1 contains no infinite upper antichains.

Immediate consequences are the known facts that a Boolean algebra B is(co)atomic iff B−1 has a cofinal subset containing no binary fork, and that aBoolean algebra with no infinite disjoint subset is already finite.

100 M. ERNE

6. Decompositions in Brouwerian Semilattices

In this final section, we have a look at situations where not only the greatestelement but all elements possess an irredundant prime decomposition.

Recall that a relatively (∨-)pseudocomplemented or Brouwerian ∨-semi-lattice is a ∨-semilattice S such that for any two elements x, y ∈ S thereis a relative ∨-pseudocomplement x\y satisfying the equivalence

x\y ≤ z ⇔ x ≤ y ∨ z .

The complete Brouwerian ∨-semilattices are exactly the coframes. From theearly work of E.T. Schmidt (see [31] or [32]), we know that

every Brouwerian ∨-semilattice S is distributive , i.e.x ≤ y ∨ z implies x = y0 ∨ z0 for suitable y0 ≤ y and z0 ≤ z.

Hence, the ∨-irreducible elements of S are exactly the ∨-primes.Each principal ideal ↓x of a Brouwerian ∨-semilattice S is clearly ∨-pseudo-

complemented (with y∗ = x \ y in ↓x); otherwise stated, S is sectionally ∨-pseudocomplemented, as mentioned in the introduction. Hence, from Proposi-tion 5 we infer the following necessary and sufficient condition for the existenceof irredundant prime decompositions:

Corollary 4. Let S be a Brouwerian ∨-semilattice. Then, an element x of Shas an irredundant prime decomposition iff the truncated principal ideal [0, x [is spoonful – in other words, iff for each a < x, there is some b ∈ [a, x [ suchthat x is ∨-prime in the interval [b, x ].

Here, half-open intervals and closed intervals are defined by[a, x [= {y ∈ L : a ≤ y < x} and [b, x ] = {y ∈ L : b ≤ y ≤ x}.

Note that ∨-primes of [0, x ] are also ∨-prime in S, because S is distributive.

Corollary 5. A coframe is spatial (dually isomorphic to a topology) wheneverall truncated principal ideals are spoonful.

The latter condition is certainly fulfilled in strongly coatomic coframes, whereeach element of any truncated principal ideal [0, x [ is dominated by a maximalone. But here a sharper result is valid (cf. Gorbunov [20]):

Corollary 6. A complete lattice is a strongly coatomic coframe if and onlyif each of its elements has an irredundant decomposition into (completely)

∨-

irreducible ∨-primes.

By relativization of Theorem 2, we obtain:

Corollary 7. An element x in a Brouwerian ∨-semilattice has a (least) finiteprime decomposition iff [0, x [ contains no infinite upper antichain.

An interesting variant of Corollary 7 holds for dual Heyting algebras, i.e. forBrouwerian ∨-semilattices that happen to be lattices.

PRIME DECOMPOSITION AND PSEUDOCOMPLEMENTATION 101

Corollary 8. If a dual Heyting algebra S has a meet-dense subset in whichall antichains are finite then S is a free ∨-semilattice over the poset P of ∨-primes. Thus, each element has a least finite prime decomposition, and S isisomorphic to the system of all finitely generated down-sets of P .

Proof. If M is a meet-dense subset containing no infinite antichains, then foreach element x, the set Mx = {m ∧ x : m ∈ M,x 6≤ m} is cofinal in [0, x [:indeed, for a < x there is an m ∈ M with a ≤ m but x 6≤ m, and it followsthat a ≤ m ∧ x < x. Clearly, Mx does not contain any infinite antichain, andin particular, [0, x [ cannot contain infinite upper antichains. Thus, Corollary7 applies. For the remaining statements, see [14]. 2

It would be of interest to know whether the converse of the last corollaryholds as well. The non-algebraic spatial coframe of all closed sets in the cofinitetopology on ω does have meet-dense subsets without infinite antichains. Forinstance, one may take the set {n \{k} : k ∈ n ∈ ω}.

For algebraic coframes, we can prove the converse of the above result indeed.A poset P is said to be principally separated (see [11], [12]) if for x 6≤ y thereexist p, q with p ≤ x, y ≤ q, p 6≤ q, and ↑ p ∪ ↓ q = P . In that case, pis ∞-prime, and q is dually ∞-prime. For the following characterizations ofprincipally separated posets, see Corollary 4.4. in [12]:

Lemma 2. For a poset P , the following are equivalent:(a) P is principally separated.(b) Each element has a join-decomposition into ∞-primes.(c) The normal completion of P is superalgebraic.(d) The normal completion of P is principally separated.

In principally separated posets, there is a close connection between ω-primes and ∞-primes:

Lemma 3. An element p of a principally separated poset P is ω-prime iff itis a directed join of ∞-primes, or equivalently, the set of all ∞-primes belowp is directed.

Proof. Let a and b be∞-primes below some ω-prime p, and define a∨ =∨{x ∈

P : a 6≤ x}. Then p 6≤ a∨ and p 6≤ b∨, where P is the disjoint union of ↑ aand ↓ a∨, and analogously for b. By ω-primeness of p, there exists a c witha∨ ≤ c and b∨ ≤ c but p 6≤ c. By join-density of the ∞-primes, there is an∞-prime d with d ≤ p but d 6≤ c. It follows that d 6≤ a∨ and d 6≤ b∨, hencea, b ≤ d ≤ p. This shows that the set of all ∞-primes below p is directed, andby principal separation, p is the join of these. On the other hand, directedjoins of ω-primes and, in particular, of ∞-primes are ω-prime (see [14]). 2

Corollary 9. In an algebraic coframe, each element has a finite prime de-composition iff there is no infinite antichain of ∞-primes.

102 M. ERNE

Proof. In view of Corollary 8, it remains to show that for an infinite antichainA of ∞-primes, the join x =

∨A has no finite prime decomposition. But that

is easy with the help of Lemma 3: if D is a prime decomposition of x theneach a ∈ A is dominated by some da ∈ D, and a 6= b implies da 6= db. (Indeed,if a, b ≤ d ∈ D then there is an ∞-prime c with a, b ≤ c ≤ d, and c in turn liesbelow some a′ ∈ A, so that a = b = c = a′, because A is an antichain.) Thus,D cannot be finite. 2

Since algebraic coframes are, up to isomorphism, just the down-set latticesof arbitrary posets, Corollary 9 may be regarded as a lattice-theoretical for-mulation of the fact that a poset has no infinite antichains iff each down-setis a finite union of ideals (see Bonnet [3] and [17, Ch.4.5]).

Generalizing a well-known notion from complete lattices to posets, we calla poset P lower continuous if for each down-directed subset D and each x ∈ Pwith D↓ ⊆ ↓x there exists a down-directed D′ ⊆ ↑D with x =

∧D′. If

P is down-complete, i.e. each down-directed subset D has a meet∧

D, thenD↓⊆ ↓x means

∧D ≤ x. The lower continuous complete lattices are just the

join-continuous lattices (cf. [19]), characterized by the identityx ∨

∧D =

∧{x ∨ d : d ∈ D}

for all elements x and all down-directed subsets D; for the dual notion ofupper continuous lattices, see [6], and for more general results on so-calledM-distributive lattices , cf. [10]. In [14], we have established the followingexistence criterion for irredundant join-decompositions into elements of anarbitrary subset R (see Chapter 6 of [6] for the restricted dual case of algebraiclattices):

Lemma 4. Let R be a subset of a lower continuous down-complete poset P . Iffor each non-zero element of P there is an essential element in R, then eachelement has an irredundant decomposition into elements of R.

While the proof of that result leans on the down-completeness assumption,there exists a ‘non-complete’ variant for Brouwerian ∨-semilattices. Note thatthe coframes, i.e. the complete Brouwerian ∨-semilattices are just the lowercontinuous and distributive complete lattices. The ω-primes that belong to adistinguished subset R have been referred to as R-primes.

Theorem 6. If for each non-minimal element of a Brouwerian ∨-semilatticethere is an essential R-prime, then each element has a (unique) irredundantjoin-decomposition into R-primes, and conversely.

Proof. Let A be the set of all essential R-primes for x (see Section 1). If xwould not be the join of A, we could choose a y with A ⊆ ↓y but x 6≤ y. Therelative ∨-pseudocomplement x \ y is the least element z whose join with ydominates x. In particular, z = x \ y satisfies x ≤ y ∨ z and z 6= 0. Hence,we find an essential R-prime p for z and a q with z 6≤ q but z ≤ p ∨ q. By

PRIME DECOMPOSITION AND PSEUDOCOMPLEMENTATION 103

definition of z, the join y∨ q cannot be above x, whereas x ≤ y∨ z ≤ p∨ y∨ q,showing that p is also essential for x. Thus, p belongs to A, whence p ≤ y.But then x ≤ p ∨ y ∨ q = y ∨ q, a contradiction. By way of contraposition, xmust be the join of the set A, and by Theorem 1, A is the unique irredundantjoin-decomposition of x into R-primes.

The converse implication follows from the fact that elements of an irredun-dant join-decomposition are essential for the decomposed element. 2

There are obvious ‘κ-versions’ generalizing several of the results in thispaper. Recall that q is a κ-spoonhandle iff it generates a κ-directed principalfilter. Calling a poset κ-spoonful if it has a cofinal subset of κ-spoonhandles,one easily verifies that P−1 is κ-spoonful iff so is the cofinal subset Q.

Theorem 7. The top element of a ∨-pseudocomplemented poset P has anirredundant join-decomposition into κ-prime elements iff P−1 is κ-spoonful.In that case, for any maximal upper antichain A of κ-spoonhandles, A∗ is theunique irredundant κ-prime decomposition of 1 in P and the unique directdecomposition into κ-prime atoms of P∗.

The proof is easily accomplished with the help of Theorem 3 and Propo-sition 4. The case κ = ∞, where ‘κ-spoonful’ means ‘coatomic’, is one ofGorbunov’s theorems about canonical decompositions [20].

As expected, there are various applications of our general results to thesituation of topological spaces – in fact, every lattice of closed sets is a coframe,hence a a complete (lower) Brouwerian lattice. By reasons of limited space,these and other applications to the decomposition theory for topological spacesand ideal systems of ordered sets are deferred to a separate note [15].

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104 M. ERNE

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Marcel Erne, Institut fur Algebra, Zahlentheorie und Diskrete Mathe-matik, Universitat Hannover, Germany

E-mail address: erne@math.uni-hannover.de

URL: http://www-ifm.math.uni-hannover.de/html/person.html?id=82