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Math. Log. Quart. 55, No. 6, 572 – 586 (2009) / DOI 10.1002/malq.200810034 

Finiteness conditions and distributive laws for 

Boolean algebras 

Marcel Erné ∗ 

Leibniz Universität Hannover, IAZD, Welfengarten 1, D-30167 Hannover, Germany 

Received 7 October 2008, revised 24 February 2009, accepted 3 March 2009 

Published online 16 November 2009 

Key words Atomistic lattice, Boolean algebra, chain condition, compact, continuous lattice, Dedekind finite, 

M-prime, M-distributive, pseudocomplemented. 

MSC (2000) 06E10, 03E25, 06B35, 06D05 

We compare diverse degrees of compactness and finiteness in Boolean algebras with each other and investigate 

the influence of weak choice principles. Our arguments rely on a discussion of infinitary distributive laws 

and generalized prime elements in Boolean algebras. In ZF set theory without choice, a Boolean algebra is 

Dedekind finite if and only if it satisfies the ascending chain condition. The Denumerable Subset Axiom (DS) 

implies finiteness of Boolean algebras with compact top, whereas the converse fails in ZF. Moreover, we derive 

from DS the atomicity of continuous Boolean algebras. Some of the results extend to more general structures 

like pseudocomplemented semilattices. 

c○ 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 

1 Introduction 

Since the pioneering work of the Tarski school in the middle of the last century, many mathematical proofs involving 

the Axiom of Choice (AC) have been revisited under the viewpoint of effectively weaker choice principles 

leading to the same conclusions. Two of these principles proved to be of particular use in quite diverse areas of 

mathematics, namely the Ultrafilter Principle 

UP 

every (proper) filter on a set may be extended to an ultrafilter, 

and the Principle of (Countable) Dependent Choices 

DC 

for any relation ϱ on a nonempty set X such that for each x∈X there is a y with xϱy, 

there exists a sequence (x n ) with x n ϱx n+1 for all n. 

Whereas Scott [39] and Tarski [41] discovered already in 1954 that the Ultrafilter Principle is equivalent to 

several topological compactness theorems and prime ideal theorems, specifically the Prime Ideal Theorem for 

Boolean algebras (PIT), it took a while until Halpern [25] was able to show that PIT does not imply the Axiom 

of Choice in ZFA (for ZF, see [26]). Later on, Pincus [37] succeeded in showing that even the combination of 

UP with DC is not enough to derive AC ; moreover, it is known from Cohen’s forcing method that the axioms UP 

and DC are independent of each other (see e.g. Jech [30]). Howard and Rubin [28] showed that PIT plus C ω ,the 

Axiom of Choice for countable families, still does not imply DC. 

Since Dedekind’s ingenious pioneering work [10] on the foundations of set theory, one knows that Dedekindfiniteness 

(which forbids equipollent proper subsets) is equivalent to the exclusion of denumerable subsets. 

Hence, the Denumerable Subset Axiom 

DS 

every infinite set contains a denumerable (= countably infinite) subset 

is tantamount to the postulate that every Dedekind-finite set is finite (i.e., equipollent to a natural number); 

∗ e-mail: erne@math.uni-hannover.de 

c○ 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Math. Log. Quart. 55, No. 6 (2009) / www.mlq-journal.org 573 

the converse is valid without assuming any choice principles. DS easily follows from C ω , which in turn is a 

consequence of DC. To show that the converse implications fail in ZF set theory and that DS does not follow 

from UP is a much more difficult task; Jech [30, pp. 81/97] has pointed out that in a certain model of set theory, 

called the basic Cohen model, the Ultrafilter Principle is valid, while the reals possess an infinite subset with no 

denumerable subset (see also [29]). For much more material concerning a large variety of choice principles and 

their dependence or independence, respectively, see Howard and Rubin [29], Jech [30, 31], the historical survey 

by Moore [36], and the fascinating book by Herrlich [27]. 

The deductive strength of the Ultrafilter Principle was demonstrated impressively by Banaschewski (see 

[1] – [4] for a selection of his earlier papers in that area). Of striking elegance is his proof of the equivalence 

between PIT and the so-called Prime Element Theorem (PET), providing prime elements in all non-trivial distributive 

complete lattices with compact top element (see [2]). A broad spectrum of applications was opened by 

the still equivalent but more flexible Separation Lemma for Frames and Quantales, saying that the complement 

of any Scott-open filter in a frame or quantale is generated as a downset by prime elements (see Banaschewski 

and Erné [5]). In view of the Prime Element Theorem, an obvious question is now: what consequences has the 

compactness of the top element for complemented frames, i.e., complete Boolean algebras? It is well-known that 

under suitable choice principles like UP or DC, such “compact Boolean frames” must already be finite; here, we 

shall see that the Denumerable Subset Axiom suffices for the same conclusion. The key is a careful proof analysis 

of the known fact that a distributive lattice satisfying both chain conditions has to be finite – indeed, it turns out 

that the same statement with “Dedekind finite” instead of “finite” can be shown in ZF set theory without any 

choice principles. From this, one easily deduces that Boolean algebras with compact top element are Dedekind 

finite. 

A closely related problem is to find seemingly weak conditions that force Boolean frames to be spatial, i.e., 

isomorphic to topologies. Clearly, the spatial Boolean frames are just the atomi(sti)c ones, being isomorphic to 

power sets. Here we shall need a few more lattice–theoretical notions. In order to avoid notational confusion and 

unnecessary completeness assumptions, we call a lattice L S-continuous [12] if each b ∈ L is the join of ⇓b, the 

intersection of all ideals having a join that dominates b , and we say L is compactly generated if each b ∈ L is a 

join of compact elements (i.e., elements c with c ≪ c, where a ≪ b means a ∈⇓b). The complete S-continuous 

lattices are the continuous lattices in the sense of Scott [24], and the complete compactly generated lattices are 

the algebraic ones [8]. By an arithmetic lattice we mean a compactly generated lattice in which the meet of any 

two compact elements is compact. It is easy to see, without any choice principles, that S-continuous Boolean 

algebras are already arithmetic (cf. [24] for the complete case); indeed, in a Boolean algebra a ≪ b and a ≪ c 

imply a≪a ≤ b∧ c (see Section 3). Since UP is equivalent to the spatiality of arithmetic frames (most easily via 

the Separation Lemma for Frames [5]), it guarantees that S-continuous complete Boolean algebras are atomistic. 

The same result, with no completeness hypothesis, was obtained in [12], using the intermediate result that in 

a Boolean algebra with enough prime ideals, ⇓⊤ is the intersection of all non-principal prime ideals, where ⊤ 

denotes the top (= greatest) element. 

On the other hand, Banaschewski and Pultr [6] deduced the atomicity of continuous Boolean algebras from 

the fact that an atomless Boolean algebra has properly descending sequences – a consequence of DC, but also, 

as our considerations will show, of the weaker axiom DS. We shall demonstrate that DS is equivalent to the 

combination of C ω

574 M. Erné: Finiteness conditions and distributive laws for Boolean algebras 

2 M-distributivity and M-generation 

Let us prepare our investigations on Boolean algebras with a brief but systematic discussion of certain infinite 

distributive laws (cf. [18]) and various kinds of irreducible, prime or compact elements in arbitrary posets. The 

point is that we have to avoid carefully any choice principles. For any subset B of a poset A, put 

↓B = {a ∈ A : a ≤ b for some b ∈ B}, B ↓ = {a ∈ A : a ≤ b for all b ∈ B}, 

↑B = {a ∈ A : a ≥ b for some b ∈ B}, B ↑ = {a ∈ A : a ≥ b for all b ∈ B}. 

While ↓B is the downset and ↑B is the upset generated by B, thesetsB ↓ and B ↑ consist of all lower and 

upper bounds of B, respectively. The set B ↑↓ is the cut generated by B. Note that a greatest element ⊤ satisfies 

⊤∈ B ↑↓ if and only if ⊤ is the join (supremum) of B, andthata ∈ B ↑↓ means a ≤ ∨ B whenever B has a 

join ∨ B. Often it has considerable advantages to work with cuts rather than joins (see e.g. [11, 15] ). As usual, 

we write ↓b for the principal ideal ↓{b} = {b} ↓ = {b} ↑↓ and ↑b for the principal filter ↑{b} = {b} ↑ = {b} ↓↑ 

generated by b ∈ A. GivenasystemM of subsets of A, we put 

M ∧ = {↓M : M ∈Mor M = {a} for some a ∈A} , 

a ≪ M b ⇔ a ∈⇓ M b = ⋂ {C ∈M ∧ : b ∈ C ↑↓ } . 

The poset A is called M-complete if each M ∈M has a join in A, M-predistributive if each b∈A is the join of 

its M-below set ⇓ M b ,andM-distributive if, in addition, A is M-complete. This terminology is justified by the 

fact that for a complete lattice, M-distributivity means 

∧ ∨ ∨⋂ { B : B ∈X} = X for all X⊆M ∧ . 

Each M-distributive lattice satisfies a ∧ ∨ M = ∨ {a ∧ b : b ∈ M} for all M ∈M.Aframe is a complete 

lattice A satisfying this equation for each M ⊆ A, andcoframes are defined dually. 

An element a of a poset is called M-irreducible if for each M ∈M, a = ∨ M implies a ∈ M, andM-prime 

or M-compact if a ≪ M a, i.e. , for each M ∈M, a ∈ M ↑↓ implies a ∈↓M. Clearly, M-prime elements are 

M-irreducible, and the converse holds at least for top elements. A poset is said to be M-primely generated 

or M-compactly generated or simply M-generated if each of its elements is a join of M-prime elements. All 

M-generated posets are M-predistributive. For a more comprehensive discussion of these topics, see Bandelt 

and Erné [7, 13, 15, 16, 18]. 

Let us list some frequently occurring examples of subset systems M and the derived notions of primeness, 

irreducibility and distributivity. 

M members M-prime M-irreducible 

A arbitrary subsets 

∨ -prime 

∨ -irreducible 

B binary subsets ∨-prime or ⊥ ∨-irreducible or ⊥ 

C chains chain-compact chain-inaccessible 

D directed subsets (D-)compact join-inaccessible 

E 1-element subsets arbitrary arbitrary 

F finite subsets ∨-prime ∨-irreducible 

M 

M-complete poset 

M-generated 

complete lattice 

M-distributive 

complete lattice 

A complete lattice superalgebraic lattice supercontinuous lattice 

B ∨-semilattice spatial coframe B-distributive complete lattice 

C chain-complete poset chain-algebraic lattice chain-continuous lattice 

D up-complete (dcpo) algebraic lattice continuous lattice 

E arbitrary poset complete lattice complete lattice 

F ∨-semilattice with ⊥ spatial coframe dual quasitopology 


www.mlq-journal.org


A few remarks are in order. If a poset has a least element ∨ ∅ , this is denoted by ⊥ . By a binary subset 

we mean a subset with one or two elements, and by a chain a nonempty linearly ordered set. The term “joininaccessible” 

[8] is suggestive but conflicts with the notion of inaccessible cardinals. 

As in topology, a top element ⊤ of a ∨-semilattice is compact if and only if ⊤ = ∨ B implies ⊤ = ∨ F for 

a finite F ⊆ B. In up-complete (but not in arbitrary) posets, compactness and D-compactness coincide, every 

compact element is chain-compact, and the reverse implication holds in ZFC, whence the chain-algebraic lattices 

are just the algebraic ones; similarly, in ZFC one can prove the equivalence of C- andD-completeness (see 

e.g. [16]). 

The question as to whether chain-continuous lattices are already continuous has been answered (in the affirmative) 

only for cardinality less than ω ω (see Markowsky [35]). 

The description of supercontinuous lattices by the complete distributive law involving choice functions (cf. Raney 

[38]) requires the full AC . Indeed, Kanai [32] showed that various choice principles are equivalent to certain 

“distributive laws” (involving choice functions) for Boolean algebras; corrections and improvements of [32] are 

given in [20]. 

The ∨-primely generated complete lattices are called spatial coframes and their duals spatial frames, because 

the latter are just the isomorphic copies of topologies. As shown in [13] and [18], the images of (super-)algebraic 

lattices are the (super-)continuous lattices, and the images of topologies under complete homomorphisms are 

the dually F-distributive complete lattices, whence these have also been baptized quasitopologies (cf. [34]). 

Induction shows that every B-distributive poset is A n -distributive for all n ≤ ω, where A n is the collection 

of all subsets with less than n elements; but it seems to be unknown whether B-distributivity is equivalent to 

F-distributivity in general. We shall see that this is the case at least for all Boolean algebras. 

Given a poset A and a collection M of subsets, let M ∨ denote the set of all members of M that possess 

a join. In the absence of M-completeness, the notions of M-predistributivity and of M-primeness have to be 

distinguished carefully from M ∨ -predistributivity and M ∨ -primeness, respectively. For example, ω × ω ,the 

square of the chain of natural numbers, is a conditionally complete, M ∨ -generated and M ∨ -distributive lattice 

for all M (because M ∨ is contained in F and all elements of the form (0,n) or (n, 0) with n ≠0are F-prime), 

but ω × ω is not M-predistributive if M contains, say, some set of the form {(x, y) :x ≤ n, y ∈ ω} .Inthis 

example we have 

⇓b = ⇓ D∨ b = ↓b but ⇓ D b = {(0, 0)} for all b ∈ ω × ω, 

whence each element is (D ∨ -)compact, but no element except (0, 0) is D-compact. 

Notice that neither of the implication arrows in the hierarchy 

M-prime ⇒ M ∨ -prime ⇒ M-irreducible 

may be reversed in general, but for top elements all three properties coincide. Fortunately, most of these discrepancies 

vanish when we are dealing with Boolean algebras, as we shall see in the next section. 

A further concept will be helpful: we say a poset A with least element ⊥ is M-founded (respectively, 

M-primely founded ) if for each b∈A with ⊥< b,thereisana with ⊥< a≪ M b (respectively, ⊥


3 Atoms and distributive laws in Boolean algebras 

An atom of a lattice or poset with least element ⊥ is minimal among all elements distinct from ⊥ . None of the 

implications 

∨ -prime ⇒ ∨-prime 

⇓ 

⇓ 

atom ⇒ ∨ -irreducible ⇒ ∨-irreducible 

may be inverted in general, but in Boolean algebras, all five properties are easily seen to be equivalent. 

Throughout this section, A denotes a Boolean algebra, ⊤ its top element, ⊥ its bottom element, and a ∗ the 

complement of a ∈A . Observe that for arbitrary elements a, b and subsets C of A, 

b ∈ C ↑↓ ⇒ a ∧ b = ∨ {a ∧ b ∧ c : c ∈ C} . 

For the proof, note that a ∧ b ∧ c ≤ d implies c ≤ d ∨ (a ∧ b) ∗ for all c ∈ C ,andifb ∈ C ↑↓ ,thenb ≤ d ∨ (a ∧ b) ∗ 

and so a ∧ b ≤ d. In particular, 

b = ∨ C ⇒ a ∧ b = ∨ {a ∧ c : c ∈ C} , 

b ∈ C ↑↓ ⇒ b = ∨ {b ∧ c : c ∈ C} . 

Henceforth, we suppose that M is a shiftable system of subsets of A. That is, we postulate that for each a ∈ A 

and M ∈M,thesets 

↓{a ∧ b : b ∈ M} and ↓{a ∨ b : b ∈ M} 

are members of M ∧ . Each system M such that f[M] ∈Mfor all M ∈Mand all isotone self-maps f : A −→ A 

is shiftable; in particular, this holds for each of the systems A,...,F listed in Section 2. Furthermore, if M is 

shiftable, then so is M ∨ , the system of all sets in M that have a join. 

Proposition 3.1 For a ∈ A \{⊥}, the following conditions are equivalent: 

(a) a is M-prime; 

(b) a is M-irreducible; 

(c) a ≪ M ⊤; 

(d) a ≪ M b for some b ∈ A. 

If B ∧ ⊆M ∧ , then the previous conditions characterize the atoms of A. 

Proof. (a)⇒ (b) and (c) ⇔ (d) hold in arbitrary posets with top. 

(b) ⇒ (c). If ⊤ = ∨ M for an M ∈M,thenwehavea = a ∧⊤ = ∨ N for N = ↓{a ∧ b : b ∈ M} ∈M ∧ , 

hence a = a ∧ b ≤ b for some b ∈ M. 

(c) ⇒ (a). If a ∈ M ↑↓ for an M ∈M,thenM ≠ ∅ (as a ≠ ⊥), and we obtain ⊤ = ∨ N for the set 

N = ↓{a ∗ ∨ b : b ∈ M} ∈M ∧ (indeed, N ⊆↓c implies a ∗ ∨ b ≤ c for all b ∈ M, hence a ∨ c = ⊤ and 

M ⊆↓c ;now,a ∈ M ↑↓ entails a ≤ c = ⊤). Thus, a≪ M ⊤ yields a ≤ a ∗ ∨ b and a ≤ b for a b ∈ M. Note that 

⊥ is not M-irreducible if ∅∈M ∧ , although ⊥≪ M ⊤ if ⊥ ≠ ⊤. 

Clearly, atoms are M-irreducible. On the other hand, if a is not an atom, say ⊥

(a) A is M-generated; 

(b) A is M-predistributive; 

(c) A is M-(primely) founded; 

(d) ⊤ isajoinofM-prime (resp. M-irreducible) elements; 

(e) ⊤ = ∨ ⇓ M ⊤. 

In each of these statements, M may be replaced with M ∨ . 

P r o o f. The implications (a) ⇒ (b) ⇒ (c) and (a) ⇒ (d) ⇒ (e) are clear. 

(c) ⇒ (e). Assume ⇓ M ⊤⊆↓b but b ≠ ⊤. Then⊥


4 Chain conditions and finiteness 

Remember that in posets, the ascending chain condition (acc) excludes properly ascending sequences, and the 

descending chain condition (dcc) has the dual meaning. The next proposition is known in ZF with choice, but 

one has to be very careful if one intends to avoid rigorously any choice principles. 

Proposition 4.1 A distributive lattice is Dedekind finite if and only if it satisfies both chain conditions (the acc 

and the dcc). 

P r o o f. Clearly, a Dedekind-finite lattice must satisfy both chain conditions. Suppose now that L is a 

Dedekind-infinite distributive lattice and choose an infinite sequence (a n ) of distinct elements in L. For each 

n ∈ ω, the sublattice L n generated by A n = {a k : k < n} is finite, and the sublattice L ω generated by 

A ω = {a k : k


Theorem 4.4 The Denumerable Subset Axiom is equivalent to each of the following four statements combined 

with C ω


P r o o f. The implications (a) ⇒ (b) ⇒ (c) are clear. 

(c) ⇒ (a). As in the proof of Theorem 4.4 we conclude from (c) that the MacNeille completion N A is a 

complete Boolean algebra with compact top element, so that by Corollary 3.2, applied to M = D, every directed 

subset of N A and so every directed subset of A has a greatest element. 

(a) ⇒ (d). D = { ∨ F : F ⊆ B, F finite} is a directed subset of A; its greatest element is the least upper 

bound of B. 

(d) ⇒ (a). If D is directed and F ⊆ D is finite with ∨ F = ∨ D, then any upper bound of F in D must be the 

greatest element of D. 

(a) ⇒ (e). If A is Noetherian, then it satisfies the acc, and by Corollary 4.2, it is Dedekind finite; moreover, all 

elements are compact. 

(e) ⇒ (c). ⊤ is ω-chain-compact by the acc and so compact by (e). 

Clearly, (f) implies (a), and if DS is valid, (e) implies (f). 

Theorem 4.8 The following statements are mutually equivalent in ZF and are derivable from either of the 

axioms DS and UP, but not conversely: 

(a) Boolean algebras with compact top are finite; 

(b) nontrivial Boolean algebras with compact top have an atom; 

(c) compact elements of Boolean algebras are finite joins of atoms; 

(d) compactly generated Boolean algebras are atomic; 

(e) S-continuous (resp. D-predistributive) Boolean algebras are atomic; 

(f) Boolean algebras whose top is a join of compact elements are atomic; 

(g) Boolean algebras whose top is the join of its way-below ideal are atomic. 

P r o o f. The implications (a) ⇒ (b) and (c) ⇒ (d) are obvious. 

(b) ⇒ (c). If c is a compact element of a Boolean algebra, then each non-minimal element of the Boolean 

algebra ↓ c is compact, too, so it contains an atom, provided (b) holds. Hence, ↓ c is atom(ist)ic and c is a finite 

join of atoms. 

(d) ⇒ (a). If the top element ⊤ and consequently each element of a Boolean algebra A is compact, then 

from (d) it follows that ⊤ is a join of atoms, a finite number of which suffice by compactness. Thus, A is a finite 

power set algebra. 

The statements (d) – (g) are equivalent by Corollary 3.4. 

By Theorem 4.4, in ZF + DS any Boolean algebra with compact top is finite. Alternatively, that conclusion 

may also be drawn from the Prime Ideal Theorem (which does not imply DS, see below): as shown in [12], the 

way-below ideal ⇓⊤ consists exactly of all finite joins of atoms if it is the intersection of prime ideals. 

It is now easy to confirm the claim that finiteness of compact Boolean algebras (which follows from UP) 

together with C ω


Corollary 4.11 For a Boolean algebra A and its top element ⊤, the following implications and equivalences 

hold in ZF: 

⊤is a finite join of atoms ⇔ A is finite 

⇓ 

⊤is compact ⇔ A is Noetherian 

⇓ 

⊤is chain-compact 

and A is chain-complete 

⇔ A has no chain without greatest element 

⇓ 

⊤is ω-chain-compact 

and A is ω C-complete 

⇔ A satisfies the acc ⇔ A is Dedekind finite 

A 

If A is complete, one has the following implications and equivalences: 

A is superalgebraic ⇔ A is supercontinuous ⇔ A is atomic 

⇓ 

A is algebraic ⇔ A is continuous ⇔ ⊤ = ∨ ⇓⊤ 

⇓ 

A is chain-algebraic ⇔ A is chain-continuous ⇔ ⊤ = ∨ ⇓ C ⊤ 

⇓ 

A is ω C-generated ⇔ A is ω C-distributive ⇔ ⊤ = ∨ ⇓ ω C ⊤ 

B 

In ZF + UP, thefirst four statements in Diagram A are equivalent, and so are the first six statements about 

complete Boolean algebras in Diagram B. 

In ZF + DS, all statements in Diagram A are equivalent, and so are all statements about complete Boolean 

algebras in Diagram B. 

The lattice of all countable or cofinite subsets of the first uncountable ordinal ω 1 is distributive (being closed 

under finite unions and intersections). Note that its top element ω 1 is ω-chain-compact but not chain-compact, 

being the union of the well-ordered chain of all smaller ordinals. 

In the atomic case, the situation becomes much more transparent: 

Theorem 4.12 For an atomic Boolean algebra A, the following conditions are equivalent in ZF: 

(a) A is finite; (a’) PA is finite; (a”) PA is Dedekind finite; 

(b) A is Noetherian; (b’) PA is Noetherian; (b”) PA satisfies the acc. 

P r o o f. The implications (a) ⇒ (b) and (a) ⇒ (a’) ⇒ (b’) ⇒ (b”) are clear, and the equivalence (a”) ⇔ (b”) has 

been established in Corollary 4.2. 

(b) ⇒ (a). Being directed, the set J of all finite joins of atoms has a greatest element b. Ifb ≠ ⊤, there would 

be an atom a ≰ b, whence b


Sometimes, Noetherian posets are defined by the condition that every nonempty subset has a maximal element; 

let us speak of strongly maximized posets in that case. Of course, every strongly maximized poset is Noetherian, 

but the converse is dubious in ZF without choice. However, in ZF + DC, one has the equivalences 

P is strongly maximized ⇔ P is Noetherian ⇔ P satisfies the acc. 

While we do not know whether all compact ( = Noetherian) Boolean algebras have to be finite in ZF, this is 

certainly the case for all strongly maximized Boolean algebras: any such Boolean algebra must be (co-)atomic 

and compact, whence the top element is a join of finitely many atoms. 

5 Booleanization and continuity 

In order to extend some of the previous results to more general structures than Boolean algebras, we need a few 

weaker complementation concepts. A bounded poset S is ∨-pseudocomplemented if for each a ∈ S there is a 

least a ∗ (called the ∨-pseudocomplement of a) with a ∨ a ∗ = ⊤, while S is ∨-semicomplemented if for each 

a>⊥ there is some c


Theorem 5.4 Let M be a system of subsets of a complete lattice A with B ∧ ⊆M ∧ .ThenA is ∨-semicomplemented 

and M-predistributive if and only if it is an atomic Boolean algebra, hence isomorphic to a power 

set. 

Proof. A B-distributive complete lattice is a coframe, hence a power set algebra if ∨-semicomplemented. 

Conversely, an atomistic Boolean algebra is ∨-semicomplemented and M-predistributive (in fact M-generated), 

because the atoms are join-dense and M-prime. 

The dual of Theorem 5.4 yields a strong spatiality criterion for quasitopologies (cf. [34] for a related result): 

Corollary 5.5 A ∧-semicomplemented quasitopology is already discrete, i.e. isomorphic to a power set. 

The completeness assumption is essential in Theorem 5.4. Consider once more the lattice L of all countable or 

cofinite subsets of ω 1 , which is distributive but neither complete nor complemented, hence not a Boolean algebra. 

However, this lattice is atomistic: the atoms are the singletons, and any such atom a is A-prime, as L \↑a = ↓a ∗ 

with a ∗ = ω 1 \ a. Hence, the lattice L is M-(primely) generated and, in particular, M-predistributive for any M. 

Moreover, L is ∨-semicomplemented (being coatomistic) but not ∨-pseudocomplemented. Nevertheless, “complete” 

may be replaced by “complemented” in Theorem 5.4. 

By the (dualized) Glivenko–Frink Theorem [23], the (∨-)skeleton or (∨-) Booleanization of any (∨-)pseudocomplemented 

(∨-)semilattice S, 

S ∗ = {a ∗ : a ∈ S} = {a ∈ S : a = a ∗∗ } 

is a Boolean algebra, with binary meet a ∧ b =(a ∗ ∨ b ∗ ) ∗ and complement a ∗ for a in S ∗ . Moreover, S ∗ is 

join-closed in S. Note also that 

(M ↑ ∩ S ∗ ) ↓ = M ↑↓ for all M ⊆ S ∗ . 

Indeed, if M ⊆ S ∗ , b ∈ (M ↑ ∩ S ∗ ) ↓ and c ∈ M ↑ ,thenc ∗∗ ∈ M ↑ ∩ S ∗ and therefore b ≤ c ∗∗ ≤ c ; thus b ∈ M ↑↓ . 

Proposition 5.6 Let S be a ∨-pseudocomplemented ∨-semilattice, M a system of subsets such that for all 

a ∈ S, M ∈Mholds ↓{a ∨ b : b ∈ M} ∈M ∧ , and put M X = {M ∈M: ∅ ≠ M ⊆ X} for X ⊆ S. Then: 

(0) a ≪ M b in S implies a ∗∗ ≪ MS∗ b ∗∗ in S ∗ . 

(1) If c ≪ M ⊤,thenA = {a ∈ S ∗ : a ≤ c} is a Boolean algebra in which every element is M A -prime; hence, 

each member of M A has a greatest element if A is M A -complete (in particular, if S is M-complete). 

(2) If S is ∨-complemented and M-founded, then S ∗ is an M S∗ -generated and hence M S∗ -predistributive 

Boolean algebra. 

P r o o f. (0) Given M ∈M S∗ and b ∈ S with b ∗∗ ∈ (M ↑ ∩ S ∗ ) ↓ = M ↑↓ ,weget⊤ = ∨ {b ∗ ∨ d : d ∈ M}, 

because b ∗ ∨d ≤ c for all d ∈ M ≠ ∅ implies c ∈ M ↑ , hence b ∗∗ ≤ c and c = b ∗ ∨c =⊤. Now,ifa ≪ M b ,there 

is a d ∈ M with a ≤ b ∗ ∨d . Furthermore, a ≤ b entails b ∗ ≤ a ∗ , and we obtain ⊤= a ∗ ∨a ≤ a ∗ ∨ b ∗ ∨d = a ∗ ∨d , 

hence a ∗∗ ≤ d , as required for a ∗∗ ≪ MS∗ b ∗∗ . 

(1) Since S ∗ is a Boolean algebra, the principal ideal A in S ∗ with top c ∗∗ is Boolean, too. As we saw in (0), 

c ≪ M ⊤ in S implies c ∗∗ ≪ MS∗ ⊤ in S ∗ , and by Corollary 3.2, it follows that all elements of A are M A -prime. 

(2) Assume ⊥< b= b ∗∗ in S ∗ and choose a ∈ S with ⊥< a≪ M b.By∨-complementation, we have a ∗


Regarding Parts (2) in Proposition 5.6 and Theorem 5.7, one might wonder if the hypothesis of ∨-complementation 

may be weakened to ∨-pseudocomplementation. The subsequent remarks will destroy that hope. 

Necessary and sufficient conditions for atomicity and for finiteness, respectively, of the skeleton of a ∨-pseudocomplemented 

semilattice or poset S have been established in [19], using the notion of upper antichains, i.e., 

subsets no two elements of which have a common upper bound in the whole poset S, and of spoonful posets, i.e., 

posets having a cofinal subset of elements that generate directed principal filters: 

Theorem 5.9 A ∨-pseudocomplemented semilattice S has a finite skeleton if and only if S ′ = S \{⊤}has 

acofinal subset containing no infinite upper antichain. Similarly, S has an atomic skeleton if and only if S ′ is 

spoonful. In ZF + DC, this is equivalent to the exclusion of binary trees no two elements of which have a common 

upper bound in S ′ . 

A straightforward verification shows that a cofinal subposet Q of a poset P is spoonful if and only if the entire 

poset P is spoonful. Applying this to P = A ∧ S ′ \{S ′ }, where A ∧ S ′ = {↓B : B ⊆ S ′ } is the downset lattice 

of S ′ = S \{⊤}, and observing that S ′ is isomorphic to the cofinal subset {S ′ \↑a : a ∈ S ′ } of P , we conclude: 

Corollary 5.10 The skeleton S ∗ is atomic if and only if the skeleton of A ∧ S ′ is atomic. 

In that way, one may associate with any ∨-pseudocomplemented semilattice S whose skeleton is not atomic 

(e.g. with any atomless Boolean algebra) a superalgebraic one, A ∧ S ′ , whose skeleton is not atomic either. This 

produces a plethora of A-generated (hence ∨-pseudocomplemented, A-distributive and weakly atomic) complete 

lattices having a non-atomic skeleton. 

6 Topological compactness for Boolean algebras 

In this final section, we compare the lattice-theoretical notion of compactness for (the top of) a Boolean algebra 

with compactness in certain intrinsic topologies. Frink [22] introduced ideals in posets as (not necessarily 

directed) 

⋂ 

subsets I such that F ↑↓ ⊆ I for each finite 

⋂ 

F ⊆ I and defined the ideal topology by taking the 

-irreducible ideals together with the dually defined -irreducible filters as a subbase for the open sets. This 

topology may be very weak in the absence of choice principles. Therefore, we define the closed ideal topology 

on a poset to be the coarsest topology making all ideals and all filters closed; assuming the Axiom of Choice, one 

concludes that the ⋂ -irreducible ideals and filters form a subbase for the closed sets. In a Boolean algebra, each 

of the following notions amounts to the same concept: 

- maximal (proper) ideal, 

- complement of an ultrafilter, 

- ⋂ -irreducible ideal, 

- complement of a ⋂ -irreducible filter, 

- prime ideal, 

- complement of a prime filter. 

Hence, the ideal topology is the coarsest topology making all prime ideals both open and closed. Thus, the Prime 

Ideal Theorem ensures that the ideal topology and the closed ideal topology coincide on Boolean algebras. In 

contrast to that coincidence, the ideal topology on [0, 1] n is the (compact) Euclidean topology, whereas the closed 

ideal topology is discrete. 

Proposition 6.1 A poset that is compact in the closed ideal topology is Noetherian. In particular, a Boolean 

algebra that is compact in the closed ideal topology must be Dedekind finite and complete. 

Proof. IfD is a directed subset of a poset with compact closed ideal topology, then {↑ d : d ∈ D}∪{↓D} 

is a system of closed sets finitely many of which always intersect. Thus, the system has a nonempty intersection, 

i.e., D has a greatest member. 


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Corollary 6.2 In ZF+UP, the following statements on a Boolean algebra A are equivalent: 

(a) A is compact in the (closed) ideal topology; 

(b) A is finite; 

(c) A is Noetherian; 

(d) the top element (and so each element) of A is compact. 

Compare this with a known result about the interval topology, the coarsest topology in which all principal 

ideals and principal filters are closed [17]: 

Proposition 6.3 The Ultrafilter Principle holds if and only if every complete (Boolean) lattice is compact in 

the interval topology. 

Corollary 6.4 For a lattice L, the following are equivalent in ZF + UP: 

(a) L and its dual are Noetherian; 

(b) the closed ideal topology on L is compact; 

(c) L is complete and the closed ideal topology coincides with the interval topology. 

For various topological conditions equivalent to the atomicity of Boolean algebras, we refer to [12]. 

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