Lattice Operators and Topologies

Abstract

Working within a complete (not necessarily atomic) Boolean algebra, we use a sublattice to define a topology on that algebra. Our operators generalize complement on a lattice which in turn abstracts the set theoretic operator. Less restricted than those of Banaschewski and Samuel, the operators exhibit some surprising behaviors. We consider properties of such lattices and their interrelations. Many of these properties are abstractions and generalizations of topological spaces. The approach is similar to that of Bachman and Cohen. It is in the spirit of Alexandroff, Frolík, and Nöbeling, although the setting is more general. Proceeding in this manner, we can handle diverse topological theorems systematically before specializing to get as corollaries as the topological results of Alexandroff, Alo and Shapiro, Dykes, Frolík, and Ramsay.

1. Introduction

We begin with ℒ, a sublattice of a complete (not necessarily atomic) Boolean algebra ℬ. If ℒ is closed under arbitrary meets, it abstracts the closed sets of a topological space. If not, we introduce a Kurotowski closure operator to define the associated topological lattice. The operators we define generalize complement on a lattice which in turn abstracts the set theoretic operator. Less restricted than those of Banaschewski [1] and Samuel [2], the operators exhibit some surprising behaviors. We consider certain properties of such lattices and the implications for the properties of one lattice from those of another, when one is a sublattice of the other. Many of these properties are abstractions and generalizations of topological spaces.

The approach is similar to that of Bachman and Cohen [3, 4]. It is in the spirit of Alexandroff [5], Frolík [6], and Nöbeling [7], although the setting is more general. We generalize a variety of filter arguments used in paved space [8, 9] and in the theory of realcompactness [10, 11]. Proceeding in this manner, we can handle diverse topological theorems systematically before specializing to get as corollaries as the topological results of [5, 8, 12–14].

Section 2 provides some background material and generates a topology on an algebra by means of a sublattice. Section 3 defines operators and topological type properties for a lattice. Section 4 examines filter and measure behavior with respect to the operators. Section 5 looks at covering properties. Section 6 investigates the relationships between two lattices.

2. Background, Terminology, and Notation

We work within a complete Boolean algebra ℬ with minimal element 𝟎 and maximal element 𝐞. The usual operators are denoted by ∨, ⋀, and ′. ℬ is not necessarily atomic; equivalently, ℬ is not necessarily completely distributive [15].

(i)𝜇,𝜇_𝑓 denote finitely additive zero-one measures on ℬ. (ii)ℒ,ℒ₁,  and ℒ₂ denote sublattices of ℬ containing 𝟎 and 𝐞.(iii)𝒜(ℒ) denotes the algebra generated by ℒ.(iv)𝒫(𝑆) is the power set of the set 𝑆. (v)The indices 𝑖,𝑗,𝑘, and 𝑛 index countable (finite or countably infinite) collections, while 𝛼,𝛽, and 𝛾 index arbitrary ones.

Definition 2.1. (i) 𝐹⊆ℒ is an ℒ-filter if and only if for all 𝑎,𝑏∈ℒ:(a)𝑎,𝑏∈𝐹⇒𝑎⋀𝑏∈𝐹,(b)𝑎∈𝐹 and 𝑎≤𝑏⇒𝑏∈𝐹.
When there is no ambiguity, we simply say that 𝐹 is a filter.
(ii) An ℒ-filter 𝐹 is a primeℒ-filter if and only if for all 𝑎,𝑏∈ℒ,𝑎∨𝑏∈𝐹⇒ either 𝑎∈𝐹 or 𝑏∈𝐹.
(iii) An ℒ-filter 𝐹 is an ℒ-ultrafilter (or ultra) if and only if 𝐹 is a maximal ℒ-filter.
(iv) A filter 𝐹 is fixed if and only if ⋀𝐹≠𝟎. Otherwise 𝐹 is free.
(v) A filter 𝐹 has cmp (countable meet property or countable intersection property) if and only if for any countable subset of 𝐹, ⋀_𝑖𝑎_𝑖≠𝟎.

Remark 2.2. It follows that (i)𝟎∉𝐹 if and only if 𝐹≠ℒ,(ii)every ultrafilter is prime. In this paper, 𝟎 is not in any filter.

Definition 2.3. A measure 𝜇 on an algebra ℬ containing ℒ is ℒ-regular if and only if for all 𝑏∈ℬ,𝜇(𝑏)=sup{𝜇(𝑎)∶𝑎∈ℒ,𝑎≤𝑏}.

Lemma 2.4. There exist one-to-one correspondences between (i)zero-one measures on ℒ and prime ℒ-filters, (ii)ℒ-regular zero-one measures on 𝒜(ℒ), and ℒ-ultrafilters [3, 4].

Thus, analogous measure theoretic results may easily be derived from our filter statements.

We now topologize ℬ by means of a sublattice ℒ. The lattice elements themselves may not be sufficient to be used as open or closed sets. However, we will generate a topology.

Definition 2.5. Let ℬ be an algebra, ℒ a sublattice of ℬ,and 𝑎,𝑏∈ℬ.(i)𝐛=⋀{𝑎∈ℒ∶𝑎≥𝑏}.(ii)𝐛^𝟎=⋁{𝑎^′∶𝑎∈ℒand𝑎^′≤𝑏}.(iii)𝑏∈ℬ is closed if and only if 𝑏=𝑏. (iv)𝜏(ℒ) denotes the set of closed elements of ℬ.

Remark 2.6. (i) 𝑏↦𝑏 is a Kurotowski closure operator.
(ii) 𝑏↦𝑏⁰ is an interior operator.
(iii) 𝜏(ℒ)={𝑏∈ℬ∶𝑏=⋀_𝛼𝑎_𝛼,𝑎_𝛼∈ℒ}.
(iv) ℒ⊆𝜏(ℒ).
(v) 𝜏(ℒ) is a lattice which is closed under arbitrary meets.

We observe that 𝜏(ℒ) is an obvious abstraction of the closed sets in a topological space.

Example 2.7. Let ℒ be the lattice of zero sets in a 𝑇₂ completely regular topological space. Then 𝜏(ℒ) is the lattice of closed sets [11].

Definition 2.8. Let ℳ⊆ℬ be a meet semilattice (i.e., a subset of ℬ closed under finite meets). Consider the generated lattice ℒ(ℳ)={⋁^𝑛₁{𝑎_𝑖∶𝑎_𝑖∈ℳ}∶𝑛∈ℕ}. All terminology remains the same except that 𝐹 is a prime ℳ-filter if and only if ⋁^𝑛₁𝑎_𝑖∈𝐹 (for 𝑎_𝑖∈ℳ) implies that one of the 𝑎_𝑖∈𝐹.

Lemma 2.9. (i) There exist one-to-one correspondences between the prime, ultra-, fixed, and free filters on ℳ and those on ℒ(ℳ).
(ii)ℳ-filters with cmp correspond with ℒ-filters with cmp.

Thus we lose no generality in “treating” ℳ like a lattice.

Example 2.10. Let ℳ={𝑏∈ℬ∶𝑏=(𝑏)⁰}. ℳ is a meet semi-lattice since 𝑎⋀𝑏≤𝑎⋀𝑏 implies that (𝑎⋀𝑏)⁰≤(𝑎⋀𝑏)⁰=(𝑎)⁰⋀(𝑏)⁰=𝑎⋀𝑏,forall𝑎,𝑏∈ℳ. “An open subset 𝐺 in a topological space is regularly open if and only if 𝐺 is the interior of its closure” [16]. Thus the regularly open sets in a topological space form a meet semi-lattice.

Table 1 summarizes the notation used in this paper.

Table 1 

Summary of notation.

3. Lattice Operators and Properties

In this section we define certain operators and lattice properties. These properties reduce to the conventional topological ones when the operator is taken to be complement.

Definition 3.1. Let ℳ be a meet semi-lattice containing 𝟎 and 𝐞. We define 𝐓 to be a one-to-one operator on ℳ such that 𝑇(𝑎∨𝑏)=𝑇(𝑎)⋀𝑇(𝑏) and 𝑇(𝐞)=𝟎. We define 𝐚^∗ = 𝑇(𝑎) and ℒ^∗={𝑎^∗∶𝑎∈ℒ}.

Proposition 3.2. (a)𝑎≤𝑏⇔𝑇(𝑎)≥𝑇(𝑏).
(b)⋁^𝑛₁𝑇(𝑎_𝑖)≤𝑇(⋀^𝑛₁𝑎_𝑖).
(c)⋁^𝑛₁𝑇(𝑎_𝑖)=𝐞⇒⋀^𝑛₁𝑎_𝑖=𝟎.
(d)𝑇(⋁^𝑛₁𝑎_𝑖)=𝟎⇒⋁^𝑛₁𝑎_𝑖=𝐞.

Proof. (a) 𝑎≤𝑏⇔𝑎∨𝑏=𝑏⇔𝑇(𝑎)⋀𝑇(𝑏)=𝑇(𝑏)⇔𝑇(𝑎)≥𝑇(𝑏),
(b) For all 𝑗,𝑎_𝑗≥⋀^𝑛₁𝑎_𝑖⇒ for all 𝑗,𝑇(𝑎_𝑗)≤𝑇(⋀^𝑛₁𝑎_𝑖)⇒⋁^𝑛₁𝑇(𝑎_𝑗)≤𝑇(⋀^𝑛₁𝑎_𝑖).
The proof of (c) and (d) follows readily.

From now on, we assume that 𝑇 is defined on a lattice ℒ. Then ℒ^∗ is a meet semi-lattice. By Lemma 2.9, we may “treat” ℒ^∗ like a lattice.

Corollary 3.3. When 𝑇 is defined on 𝜏(ℒ), one has the following. (a)⋁_𝛼𝑇(𝑎_𝛼)≤𝑇(⋀_𝛼𝑎_𝛼). (b)⋁_𝛼𝑇(𝑎_𝛼)=𝐞⇒⋀_𝛼𝑎_𝛼=𝟎.

Example 3.4. Let 𝑆={1,2,3,4} and ℬ=𝒫(𝑆) (the power set of 𝑆) with set union and intersection as the join and meet operations. Let ℒ={∅,𝑎,𝑏,𝑆} with 𝑎={1,2} and 𝑏={3,4} as in Figure 1. The operator 𝑇 is defined by 𝑇(𝑆)=∅,𝑇(𝑎)={4},𝑇(𝑏)={2}, and 𝑇(∅)={2,4}. We have 𝑇(𝑎∨𝑏)=𝑇(𝑎)⋀𝑇(𝑏) and 𝑇(𝑆)=∅. In addition 𝑇(𝑎⋀𝑏)=𝑇(𝑎)∨𝑇(𝑏) and 𝑇(𝑎)⋀𝑎=∅. Note that, unlike in [2], 𝑇(𝑎) is not the maximal element disjoint from 𝑎, and although ℒ is complemented (that is, 𝑎∈ℒ⇒𝑎^′∈ℒ),𝑇(𝑎)≠𝑎^′.

Figure 1 

ℒ in Example 3.4.

We now define various properties for ℒ. They generalize some of the definitions in point set topology, reducing to the conventional properties when ℬ is the power set of a set 𝑋 and 𝑇 is complement.

Definition 3.5. (i) ℒ is compact if and only if every ℒ-filter is fixed.
(ii) ℒ is ℵ₀-compact if and only if every ℒ-filter has cmp.
(iii) ℒ is an I-lattice (P-lattice) if and only if every (prime) ℒ-filter with cmp is contained in an ultrafilter with cmp.
(iv) ℒ is an R-lattice if and only if every filter which contains a fixed prime filter is also fixed.
(v) A topological space is an I-space if and only if the lattice of closed sets is an I-lattice [17].

Proposition 3.6. As an immediate consequence of Definition 3.5, one has the following.(i)ℒ is compact ⇒ℒ is ℵ₀-compact ⇒ℒ is an I-lattice ⇒ℒ is a 𝑃-lattice.(ii)The following are equivalent: (a)ℒ is compact (ℵ₀-compact),(b)every prime filter is fixed (has cmp),(c)every ultrafilter is fixed (has cmp).

Definition 3.7. (i) ℒ is ℵ_𝟎-paracompact if and only if whenever there exists {𝑎_𝑛}⊆ℒwith𝑎_𝑛↓𝟎, there exists {𝑏_𝑛}⊆ℒsuch that 𝑎_𝑛≤𝑏^∗_𝑛 and 𝑏^∗_𝑛↓𝟎.
(ii) When ℒ⊆𝜏(ℒ^∗), we say that ℒ is perfect.

Proposition 3.8. Every perfect lattice is ℵ₀-paracompact.

Proof. Assume ℒ is perfect and 𝑎_𝑛↓𝟎. For each 𝑎_𝑛, there exists {𝑏_𝑛𝛼}⊆ℒ such that 𝑎_𝑛=⋀_𝛼𝑏^∗_𝑛𝛼. Let 𝑐^∗_𝑛=⋀{𝑏^∗_𝑘𝛼∶𝑘≤𝑛}. Then 𝑐^∗_𝑛≥𝑎_𝑛 and 𝑐^∗_𝑛↓𝟎.

Example 3.9. The zero sets in a topological space are perfect (i.e., complement generated in the sense of [11]) and thus ℵ₀-paracompact.

Definition 3.10. Let 𝑎,𝑎_𝑖,𝑏∈ℬ,and𝑓,𝑓_𝑖,𝑔,𝑔_𝑖∈ℒ. Then the following are given. (i)ℒ is 𝐓₁ if and only if for all 𝑎₁≰𝑎₂ there exists 𝑔^∗∈ℒ^∗ such that 𝑎₂≤𝑔^∗ but 𝑎₁≰𝑔^∗.(ii)ℒ is Hausdorff if and only if for all 𝑎₁,𝑎₂≠𝟎 with 𝑎₁⋀𝑎₂=𝟎 there exist 𝑔^∗₁,𝑔^∗₂∈ℒ^∗ such that 𝑎₁⋀𝑔^∗₁≠𝟎,𝑎₂⋀𝑔^∗₂≠𝟎,and𝑔^∗₁⋀𝑔^∗₂=𝟎.(iii)ℒ is regular if and only if for all 𝑏∈ℬ,𝑏≠𝟎,𝑓∈ℒ,𝑏⋀𝑓=𝟎 there exist 𝑔^∗₁,𝑔^∗₂∈ℒ^∗ such that 𝑏⋀𝑔^∗₁≠𝟎,𝑓≤𝑔^∗₂, and 𝑔^∗₁⋀𝑔^∗₂=𝟎.

The following proposition provides an example.

Proposition 3.11. Let ℬ be the power set of a topological space 𝑋, let ℒ be the lattice of closed sets in 𝑋, and let 𝑇 be complement. 𝑋 is a 𝑇₁ (Hausdorff, regular) space if and only if ℒ is 𝑇₁ (Hausdorff, regular).

Proof. Let ℒ be the lattice of closed sets in 𝑋. Suppose ℒ is a 𝑇₁ lattice. Let 𝑎₁≠𝑎₂ be atoms in ℬ. Since 𝑎₁≰𝑎₂, there exists 𝑔^′₂∈ℒ^′ such that 𝑎₂<𝑔^′₂ but 𝑎₁≰𝑔^′₂. By symmetry, there exists 𝑔^′₁∈ℒ^′ such that 𝑎₁≤𝑔^′₁ but 𝑎₂≰𝑔^′₁. Thus 𝑋 is a 𝑇₁ space.
Now suppose 𝑋 is a 𝑇₁ topological space. Let 𝑏,𝑐∈ℬ,𝑏≰𝑐. Then there exists an atom 𝑎≤𝑏 but 𝑎≰𝑐. Thus for all atoms 𝑎_𝛼≤𝑐,𝑎≠𝑎_𝛼, and there exists 𝑔^′_𝛼∈ℒ^′ such that 𝑎_𝛼≤𝑔^′_𝛼 but 𝑎≰𝑔^′_𝛼. Let 𝑔^′=⋁_𝛼𝑔^′_𝛼. Then 𝑐≤𝑔^′,𝑏≰𝑔^′, and thus ℒ is 𝑇₁.
The proofs for Hausdorff and regular are similar.

Proposition 3.12. Suppose 𝑇(𝑎)⋀𝑎=𝟎 and ℒ is regular. If 𝐹₁ is prime and 𝐹₁⊆𝐹₂, then ⋀𝐹₁=⋀𝐹₂.

Proof. Suppose there exists 𝐹₁⊆𝐹₂such that ⋀𝐹₁≠⋀𝐹₂. Let 𝑎=⋀𝐹₁.𝑎≰⋀𝐹₂ implies there exists 𝑓∈𝐹₂with𝑎≰𝑓. But then 𝑏=𝑎⋀𝑓^′≠𝟎 and 𝑏⋀𝑓=𝟎. By regularity, there exist 𝑐^∗₁,𝑐^∗₂∈ℒ^∗ such that 𝑏⋀𝑐^∗₁≠𝟎,𝑓≤𝑐^∗₂, and 𝑐^∗₁⋀𝑐^∗₂=𝟎. Now 𝑓≤𝑐^∗₂ implies that 𝑐₂∉𝐹₂⊇𝐹₁. From 𝑏⋀𝑐^∗₁≠𝟎, it follows that 𝑎≰𝑐₁ and thus 𝑐₁∉𝐹₁. But 𝑐^∗₁⋀𝑐^∗₂=𝟎 implies that 𝑐₁∨𝑐₂=𝐞, and thus 𝐹₁ is not prime.

Example 3.13. The closed sets in a regular topological space form an 𝑅-lattice.

Definition 3.14. ℒ is normal if and only if for all 𝑓₁,𝑓₂∈ℒ,𝑓₁⋀𝑓₂=𝟎 there exist 𝑔^∗₁,𝑔^∗₂∈ℒ^∗ such that 𝑓₁≤𝑔^∗₁, 𝑓₂≤𝑔^∗₂, and 𝑔^∗₁⋀𝑔^∗₂=𝟎.

The following proposition demonstrates an example of an application.

Proposition 3.15. Let 𝑋 be a topological space. Let 𝑇 be complement and let ℒ^′ be the lattice of open sets. Then the following are equivalent.(a)ℒ^′ is normal.(b)𝑎^′,𝑏^′∈ℒ^′ and 𝑎^′⋀𝑏^′=𝟎⇒𝑎^′⋀𝑏^′=𝟎.(c)𝑎^′∈ℒ^′⇒𝑎^′∈ℒ^′ (i.e., 𝑋 is extremally disconnected).

Proof. Consider the following. (i) (a) implies (b): let 𝑎^′,𝑏^′∈ℒ^′,𝑎^′⋀𝑏^′=𝟎. Then there exist 𝑐,𝑑∈ℒ such that 𝑎^′≤𝑐,𝑏^′≤𝑑,and𝑐⋀𝑑=𝟎 (by normality). But 𝑎^′≤𝑐 and 𝑏^′≤𝑑 implies that 𝑎^′⋀𝑏^′=𝟎.
(ii) (b) implies (c): let 𝑎^′∈ℒ^′. Let 𝑏=𝑎^′ so that 𝑎^′⋀𝑏^′=𝟎. Then 𝑎^′⋀𝑏^′=𝟎 (by hypothesis), so that 𝑏⋀𝑏^′=𝟎. Then 𝑏≤(𝑏^′)^′=𝑏⁰, so that 𝑏=𝑏⁰and𝑏∈ℒ^′. Therefore 𝑎^′∈ℒ^′ by definition of 𝑏.
(iii) (c) implies (b): let 𝑎^′,𝑏^′∈ℒ^′,𝑎^′⋀𝑏^′=𝟎, so that 𝑎^′⋀𝑏^′=𝟎. Since 𝑎^′∈ℒ^′, then 𝑎^′⋀𝑏^′=𝟎.
(iv) (b) implies (a): 𝑎^′,𝑏^′ are elements of ℒ.

Proposition 3.16. Let 𝑇(𝑎)⋀𝑎=𝟎. If ℒ is normal and 𝐹 is a prime ℒ-filter contained in two ℒ-ultrafilters 𝐺 and 𝐻, then 𝐺=𝐻.

Proof. Let 𝐺 and 𝐻 be two distinct ultrafilters and let 𝐹⊆𝐺∩𝐻. Then there exist 𝑔∈𝐺,ℎ∈𝐻with 𝑔⋀ℎ=𝟎. By normality, there exist 𝑎^∗,𝑏^∗∈ℒ^∗ such that 𝑔≤𝑎^∗,ℎ≤𝑏^∗,and𝑎^∗⋀𝑏^∗=𝟎. But 𝑔⋀𝑎=𝟎 implies that 𝑎∉𝐺, and ℎ⋀𝑏=𝟎 implies that 𝑏∉𝐻, so that 𝑎,𝑏∉𝐺∩𝐻⊇𝐹, that is, 𝑎,𝑏∉𝐹. But 𝑎∨𝑏=𝐞 implies that 𝐹 is not prime.

Definition 3.17. ℒ is ℵ₀-normal if and only if it is normal and ℵ₀-paracompact.

Example 3.18. A normal topological space is countably paracompact if and only if the lattice of closed sets is ℵ₀-paracompact [18]. Willard [16] calls such a space binormal.

Proposition 3.19. Let 𝑇 be complement in ℬ. If ℒ is ℵ₀-normal and 𝐹₁ is a prime filter with cmp, then 𝐹₁⊆𝐹₂ implies that 𝐹₂ has cmp.

Proof. Let 𝐹₁⊆𝐹₂ where 𝐹₁ is a prime ℒ-filter and 𝐹₂ is an ℒ-filter without cmp. Then there exists {𝑓_𝑘}⊆𝐹₂ such that ⋀_𝑘{𝑓_𝑘}=𝟎. Let 𝑔_𝑛=⋀^𝑛₁𝑓_𝑘.{𝑔_𝑛}⊆𝐹₂ and 𝑔_𝑛↓𝟎. Thus there exist {𝑏_𝑛}⊆ℒ such that 𝑔_𝑛≤𝑏^′_𝑛 and 𝑏^′_𝑛↓𝟎. Now 𝑔_𝑛⋀𝑏_𝑛=𝟎, so there exist two sequences {𝑐_𝑛},{𝑑_𝑛} such that 𝑔_𝑛≤𝑐^′_𝑛,𝑏_𝑛≤𝑑^′_𝑛,𝑐_𝑛⋀𝑑_𝑛=𝟎, for all 𝑛. Since 𝑐_𝑛∨𝑑_𝑛=𝐞and 𝐹₁ is prime, we have 𝑐_𝑛∈𝐹₁ or 𝑑_𝑛∈𝐹₁, for all 𝑛. But 𝑔_𝑛≤𝑐^′_𝑛 implies that 𝑐_𝑛∉𝐹₁, so𝑑_𝑛∈𝐹₁, for all 𝑛. And since 𝑏^′_𝑛≥𝑑_𝑛 implies that 𝑑_𝑛↓𝟎, we have that 𝐹₁ does not have cmp.

Remark 3.20. If ℒ is a normal lattice that has the stronger property that whenever {𝑎_𝑛}⊆ℒ such that ⋁_𝑛𝑎^∗_𝑛=𝐞 there exists {𝑏_𝑛}⊆ℒ such that ⋁_𝑛𝑏_𝑛=𝐞 and for all 𝑛,𝑎^∗_𝑛≥𝑏_𝑛, then we need only to assume that 𝑇(𝑎)⋀𝑎=𝟎.

Corollary 3.21. Let 𝑇 be complement in ℬ. If ℒ is ℵ₀-normal, then ℒ is a 𝑃-lattice.

4. Behavior of Filters and Measures Under 𝑇

In this section we look at the behavior of filters and measures with respect to 𝑇. It is interesting to see an example where 𝑇 does not “behave as nicely” as complement.

Definition 4.1. Let 𝐷⊆ℒ. 𝐃={𝑎^∗∈ℒ^∗∶𝑎∉𝐷}.

Proposition 4.2. Let 𝐹 be a prime ℒ-filter. 𝐹 is a prime ℒ^∗-filter.

Proof. (a) 𝟎∉𝐹 since 𝐞∈𝐹.
(b) 𝑎^∗≤𝑏^∗,𝑎^∗∈𝐹⇒𝑎≥𝑏 and 𝑎∉𝐹⇒𝑏∉𝐹⇒𝑏^∗∈𝐹.
(c) Let 𝑎^∗,𝑏^∗∈𝐹, so that 𝑎,𝑏∉𝐹; equivalently, 𝑎∨𝑏∉𝐹. But then (𝑎∨𝑏)^∗∈𝐹, and so 𝑎^∗⋀𝑏^∗∈𝐹. Thus (by (a), (b), and (c)), 𝐹 is a filter.
(d) Let 𝑎^∗,𝑏^∗∈ℒ^∗,𝑎^∗∨𝑏^∗≥𝑐^∗∈𝐹. (𝑎⋀𝑏)^∗≥𝑎^∗∨𝑏^∗≥𝑐^∗∈𝐹⇒𝑎⋀𝑏≤𝑐∉𝐹⇒𝑎⋀𝑏∉𝐹⇒𝑎∉𝐹 or 𝑏∉𝐹⇒𝑎^∗∈𝐹 or 𝑏^∗∈𝐹. Thus 𝐹 is prime.

Definition 4.3. A filter 𝐹 is coultra if and only if 𝐹 is ℒ^∗-ultra. 𝜇_𝐹 is coregular if and only if 𝐹 is coultra.

Proposition 4.4. 𝐹 is ℒ^∗-ultra if and only if 𝑎∈𝐹 is equivalent to 𝑎^∗≤(𝑏^∗)^′ for some 𝑏^∗∈𝐹 (i.e., 𝑎^∗∉𝐹⇔ there exists 𝑏^∗∈𝐹 such that 𝑎^∗≤(𝑏^∗)^′).

Remark 4.5. Proposition 4.4 generalizes a theorem of [12] which we get by taking 𝑇 to be complement.

As demonstrated by the following example, we can associate measures with the prime filters on ℒ as usual, but they may lack some of the properties to which we are accustomed.

Example 4.6. Let 𝑆={1,2,3,4}. Let ℬ=𝒫(𝑆), with set union and intersection as the join and meet operations. Figure 2 defines the sublattice ℒ.

Figure 2 

ℒ in Example 4.6.

Define 𝑇 on ℒ as follows:

𝑇(𝐞)=𝟎,𝑇(𝟎)=𝐞,𝑇(𝑎)=𝑔,𝑇(𝑔)=𝑎,𝑇(𝑏)=ℎ,𝑇(ℎ)=𝑏,𝑇(𝑐)=𝑐,𝑇(𝑑)=𝑑,𝑇(𝑓)=𝑓. (4.1)

Incidentally, ℒ=ℒ^∗ and for all 𝑎∈ℒ,𝑇(𝑇(𝑎))=𝑎. However 𝑎⋀𝑇(𝑎) does not necessarily = 𝟎.

Now 𝑇 can be extended to all of 𝒜(ℒ)=ℬ by defining 𝑇(𝑎∨𝑏)=𝑇(𝑎)⋀𝑇(𝑏),𝑇(𝑎⋀𝑏)=𝑇(𝑎)∨𝑇(𝑏),and𝑇(𝑎^′)=𝑇(𝑎)^′.

Let 𝐹=𝐹_𝑓={𝑥∈ℒ∶𝑥≥𝑓}={𝐞,𝑏,𝑓}.𝐹={𝑎^∗,𝑑^∗,ℎ^∗,𝑐^∗,𝑔^∗,𝟎^∗}={𝑔,𝑑,𝑏,𝑐,𝑎,𝐞}=𝐹_𝑔. Since 𝐹 is an ℒ^∗-ultrafilter, 𝐹 is a coultra filter. 𝐹 is a prime filter so we have the associated measure 𝜇=𝜇_𝐹=𝜇_𝑓, where

𝜇(𝑥)=1if𝑥≥𝑓,0otherwise. (4.2)

Now consider that {3}∈𝒜(ℒ).𝜇({2,3})=1,𝜇({2})=0, so that 𝜇({3})=𝜇({2,3})−𝜇({2})=1. But {3}≱ any element of ℒ^∗ whose measure is one, so 𝜇 is not ℒ^∗-regular even though it is ℒ-coregular.

Note that

𝜇(𝑎)=0and𝜇(𝑎^∗)=𝜇(𝑔)=0,𝜇(𝑓)=1and𝜇(𝑓^∗)=𝜇(𝑓)=1,𝜇(ℎ)=0but𝜇(ℎ^∗)=𝜇(𝑏)=1,𝜇(𝑏)=1but𝜇(𝑏^∗)=𝜇(ℎ)=0.

Thus, 𝑇 is not necessarily measure inverting (𝜇(𝑇(𝑎))=1−𝜇(𝑎)∀𝜇), preserving, increasing, or decreasing.

Remark 4.7. If 𝑇(𝑎)⋀𝑎=𝟎 and 𝑇 is measure inverting, then every ℒ-coregular measure is ℒ^∗-regular. These concepts are equivalent when 𝑇 is complement.

From now on we assume that 𝑇(𝑎⋀𝑏)=𝑇(𝑎)∨𝑇(𝑏). Now ℒ^∗ is a lattice.

Example 4.8. See Examples 4.6 and 3.4.

Proposition 4.9. 𝐹 is a prime ℒ-filter if 𝐹 is a prime ℒ^∗-filter. (Please see Definition 4.1 and Proposition 4.2 for the converse.)

Proof. (a) 𝟎∉𝐹 since 𝐞∈𝐹.
(b) 𝑎,𝑏∈𝐹⇒𝑎^∗,𝑏^∗∉𝐹⇒𝑎^∗∨𝑏^∗∉𝐹⇒(𝑎⋀𝑏)^∗∉𝐹⇒𝑎⋀𝑏∈𝐹.
(c) 𝑎≤𝑏and𝑎∈𝐹⇒𝑏^∗≤𝑎^∗and𝑎^∗∉𝐹⇒𝑏^∗∉𝐹⇒𝑏∈𝐹.
(d) 𝑎∨𝑏∈𝐹⇒(𝑎∨𝑏)^∗∉𝐹⇒𝑎^∗⋀𝑏^∗∉𝐹⇒𝑎^∗∉𝐹or𝑏^∗∉𝐹⇒𝑎∈𝐹or𝑏∈𝐹.

Corollary 4.10. (a)𝐹 is a prime ℒ-filter if and only if 𝐹 is a prime ℒ^∗-filter.
(b)𝐹 is a prime filter if 𝐹 is a coultra filter.

5. Covering Properties

In this section we define some covering properties for ℒ and show that they are analogous to the topological ones. In particular, when 𝑇 is taken to be complement, we get topological results as corollaries.

Definition 5.1. ℒ is comax compact if and only if every coultra filter is fixed. ℒ is comax ℵ₀-compact if and only if every coultra filter has cmp.

Proposition 5.2. Every comax compact 𝑅-lattice is compact.

Proof. Let 𝐹 be a prime ℒ-filter. Form 𝐹 and extend it to 𝐺, an ℒ^∗-ultrafilter. 𝐺 is a prime ℒ-filter and fixed. 𝐺⊆𝐹 implies that 𝐹 is fixed since ℒ is an 𝑅-lattice. Thus ℒ is compact by Proposition 3.6.

Corollary 5.3. If 𝑇(𝑎)⋀𝑎=𝟎 and ℒ is a comax compact regular lattice, then ℒ is compact.

Definition 5.4. ℒ is (prime, max, comax) complete if and only if every (prime, ultra-, coultra) filter with cmp is fixed.

We have the implications in Figure 3.

Figure 3 

Lattice Implications.

Proposition 5.5. If ℒ is (comax-)ℵ₀-compact, then it is (comax) complete if and only if it is (comax) compact.

Proposition 5.6. If ℒ is an R-lattice, then ℒ is prime complete if and only if it is comax complete.

Proof. In the proof of Proposition 5.2, let 𝐹 have cmp.

Proposition 5.7. If ℒ is a max complete I-lattice, then ℒ is complete.

Proof. Let 𝐹 be a filter with cmp. Since ℒ is an I-lattice, 𝐹 can be extended to 𝐺, an ultrafilter with cmp. 𝐺 is fixed because ℒ is max complete. Hence 𝐹 is fixed and ℒ is complete.

Corollary 5.8. If L is a max complete P-lattice, then ℒ is prime complete.

Proof. In the proof of Proposition 5.7, take 𝐹 to be a prime filter.

Remark 5.9. When 𝑇 is defined on 𝜏(ℒ) (as when ℒ=𝜏(ℒ)) and 𝑇(⋀_𝛼𝑎_𝛼)=⋁_𝛼𝑇(𝑎_𝛼)∀𝑎_𝛼∈ℒ, our definitions coincide with the conventional topological ones. (See Examples 3.4 and 4.6.) In particular 𝑇 may be taken to be complement.

Proposition 5.10. Let 𝑇 be defined on 𝜏(ℒ) with 𝑇(⋀_𝛼𝑎_𝛼)=⋁_𝛼𝑇(𝑎_𝛼) for all 𝑎_𝛼∈ℒ.(a)ℒ is compact if and only if 𝐞=⋁_𝛼𝑎^∗_𝛼⇒𝐞=⋁^𝑛₁𝑎^∗_𝛼𝑖. (b)ℒ is complete if and only if 𝐞=⋁_𝛼𝑎^∗_𝛼⇒𝐞=⋁^∞₁𝑎^∗_𝛼𝑖. (c)ℒ is ℵ₀-compact if and only if 𝐞=⋁^∞₁𝑎^∗_𝑖⇒𝐞=⋁^𝑛_𝑗=1𝑎^∗_𝑖𝑗.

Proof. We will prove only part (b). Parts (a) and (c) have similar proofs.
Suppose the condition holds. Let 𝐹={𝑓_𝛼} be a free ℒ-filter. Then ⋀_𝛼𝑓_𝛼=𝟎 implies that ⋁_𝛼𝑓^∗_𝛼=𝐞, so that there exists {𝑓_𝛼𝑖}⊆{𝑓_𝛼} such that ⋁_𝑖𝑓^∗_𝛼𝑖=𝐞. But then ⋀_𝑖𝑓^∗_𝛼=𝟎 and 𝐹 does not have cmp, so ℒ is complete.
Conversely, suppose that the condition does not hold. Then there exists {𝑎_𝛼} such that ⋁_𝛼𝑎^∗_𝛼=𝐞 but ⋁_𝑖𝑎^∗_𝛼𝑖≠𝐞 for any countable subset. Then 𝟎≠⋀_𝑖𝑎_𝛼𝑖 and {𝑎_𝛼} is a subbase for a filter 𝐹. 𝐹 has cmp since {𝑓_𝑗}⊆𝐹 implies that ⋀_𝑗𝑓_𝑗≥⋀_𝑗𝑎_𝛼𝑗>𝟎.𝐹 is free since 𝐞=⋁_𝛼𝑎^∗_𝛼 implies that ⋀_𝛼𝑎_𝛼=𝟎, and therefore, ℒ is not complete.

Corollary 5.11. Let 𝑋 be a topological space. Take ℒ to be the closed sets and take 𝑇 to be complement. Then 𝑋 is compact (Lindelöf, countably compact) if and only if ℒ is compact (complete, ℵ₀-compact).

Corollary 5.12. 𝑋 is a realcompact I-space if and only if it is a Lindelöf space [14].

Corollary 5.13. If 𝑋 is regular, countably paracompact, and almost realcompact, then 𝑋 is realcompact [6].

6. Lattice Interrelations

In this section we investigate the implications between the properties of two lattices when one is a sublattice of the other.

Proposition 6.1. Let ℒ₁⊆ℒ₂⊆𝜏(ℒ₁)(a)If ℒ₁ is complete, then ℒ₂ is complete. (b)If ℒ₁ is prime complete, then ℒ₂ is prime complete. (c)If ℒ₁ is a P-lattice, then that ℒ₁ is max complete implies that ℒ₂ is prime complete.

Proof. (a) Let 𝐹 be an ℒ₂-filter with cmp, 𝐺=𝐹∩ℒ₁, and 𝑏∈𝐹. There exists {𝑎_𝛼}⊆ℒ₁ such that 𝑏=⋀_𝛼𝑎_𝛼. As 𝑏≤𝑎_𝛼,for all 𝛼, we have 𝑎_𝛼∈𝐹∩ℒ₁(which=𝐺),for all 𝛼. Now let {𝑏_𝛽}⊆𝐹.⋀_𝛽𝑏_𝛽=⋀_𝛼,𝛽𝑎_𝛼,𝛽≠𝟎, since 𝐺 is fixed. Thus 𝐹 is fixed and ℒ₂ is complete.
(b) Let 𝐹 in (a) be prime. Then 𝐺 is prime.
(c) Since ℒ₁ is a P-lattice, 𝐺 may be extended to an ℒ₁-ultrafilter 𝐻 with cmp. Since ℒ₁ is max complete, 𝐻 is fixed, and hence 𝐺 and 𝐹 are fixed.

Corollary 6.2. Let ℒ₁⊆ℒ₂⊆𝜏(ℒ₁) and let ℒ₁ be a P-lattice. If ℒ₁ is max complete, then ℒ₂ is max and comax complete.

Corollary 6.3. Let ℒ₁⊆ℒ₂⊆𝜏(ℒ₁) and let ℒ₁ be ℵ₀-normal. If ℒ₁ is max complete, then ℒ₂ is max and comax complete.

Corollary 6.4. If 𝑋 is a normal, countably paracompact space, then 𝑍-replete implies 𝐹-replete and realcompact implies 𝛼-complete [13].

The following proposition generalizes two results of Alexandroff [5].

Proposition 6.5. Let ℒ₁⊆ℒ₂⊆𝜏(ℒ₁). (a)If ℒ₁ is compact, then ℒ₂ is compact. (b)If ℒ₁ is compact and normal, then ℒ₂ is normal.

Proof. (a) Let 𝐹 be an ℒ₂-filter. Let 𝐺=𝐹∩ℒ₁. The proof follows as in Proposition 6.1.
(b) Let 𝑎₁,𝑎₂∈ℒ₂,𝑎₁⋀𝑎₂=𝟎, where 𝑎₁=⋀_𝛽𝑏_𝛽,𝑎₂=⋀_𝛾𝑐_𝛾, and of course 𝑏_𝛽,𝑐_𝛾∈ℒ₁.𝑎₁≤𝑏_𝛽and𝑎₂≤𝑐_𝛾,for all 𝛽,𝛾. Now there must exist 𝑏_𝛽0,𝑐_𝛾0 such that 𝑏_𝛽0⋀𝑐_𝛾0=𝟎. (If not, {𝑏_𝛽,𝑐_𝛾} would form a subbase for a free filter in a compact space.) By normality of ℒ₁, there exist 𝑑^∗₁,𝑑^∗₂∈ℒ^∗₁ such that 𝑑^∗₁≥𝑏_𝛽0,𝑑^∗₂≥𝑐_𝛾0, and 𝑑^∗₁⋀𝑑^∗₂=𝟎. But then 𝑑^∗₁≥𝑎₁ and 𝑑^∗₂≥𝑎₂ and so ℒ₂ is normal.

Proposition 6.6. Let 𝛿(ℒ) denote the smallest set containing ℒ and closed under countable meets. Let ℒ₁⊆ℒ₂⊆𝛿(ℒ₁). If ℒ₁ is ℵ₀-compact, then ℒ₂ is ℵ₀-compact.

Proposition 6.7. Let ℒ₁⊆ℒ₂. If ℒ₂ is complete, then ℒ₁ is complete.

Proof. Let 𝐹 be an ℒ₁-filter with cmp. Let 𝐺={𝑎∈ℒ₂∶𝑎≥𝑓forsome𝑓∈𝐹}.𝐺 has cmp since ⋀𝑔_𝑖≥⋀𝑓_𝑖>𝟎,𝑔_𝑖∈𝐺,  and 𝑓_𝑖∈𝐹.ℒ₂ is complete so 𝐺 is fixed. Thus 𝐹 is fixed and ℒ₁ is complete.

Corollary 6.8. Letℒ₁⊆ℒ₂ where ℒ₂ is an I-lattice.(a)If ℒ₂ is max complete, then ℒ₁ is complete. (b)If ℒ₂ is a comax complete R-lattice, then ℒ₁ is complete.

Proof. (a) Since ℒ₂ is an I-lattice, ℒ₂ that is max complete implies that ℒ₂ is complete, which implies by Proposition 6.7 that ℒ₁ is complete.
(b) Since ℒ₂ is an R-lattice, its being comax complete implies that it is prime complete. Since ℒ₂ is an I-lattice, ℒ₂ is complete and thus ℒ₁ is complete by Proposition 6.7.

Definition 6.9. ℒ₂ is an ℒ₁-P-lattice if and only if every ℒ₁ prime filter with cmp is contained in an ℒ₂ ultrafilter with cmp.

Definition 6.10. (i) ℒ₂ is ℒ₁-normal if and only if for all 𝑓₁,𝑓₂∈ℒ₂ with 𝑓₁⋀𝑓₂=𝟎 there exist 𝑔^∗₁,𝑔^∗₂∈ℒ₁^∗ such that 𝑓₁≤𝑔^∗₁,𝑓₂≤𝑔^∗₂, and 𝑔^∗₁⋀𝑔^∗₂=𝟎.
(ii) ℒ₂ is ℒ₁-ℵ₀-paracompact if and only if for each {𝑎_𝑛}⊆ℒ₂ such that 𝑎_𝑛↓𝟎 there exists a sequence {𝑏_𝑛}⊆ℒ₁ such that 𝑎_𝑛≤𝑏^∗_𝑛 and 𝑏^∗_𝑛↓𝟎.
(iii) ℒ₂ is ℒ₁-ℵ₀-normal if and only if ℒ₂ is ℒ₁-normal and ℒ₁-ℵ₀-paracompact.

Proposition 6.11. Let 𝑇 be complement. If ℒ₂ is ℒ₁-ℵ₀-normal and 𝐹₁ is a prime ℒ₁-filter with cmp contained in an ℒ₂-filter 𝐹₂, then 𝐹₂ has cmp.

Proof. Similar to the proof of Proposition 3.19.

Corollary 6.12. Let 𝑇 be complement. (a)That ℒ₂ is ℒ₁-ℵ₀-normal implies that ℒ₂ is an ℒ₁-P-lattice. (b)If ℒ₂ is ℵ₀-paracompact and ℒ₁ separates ℒ₂, then ℒ₂ is ℒ₁-ℵ₀-paracompact.

Corollary 6.13. (a) Let ℒ₂ be an ℒ₁-P-lattice. That ℒ₂ is max complete implies that ℒ₁ is prime complete.
(b) If in addition ℒ₂ is an R-lattice, then that ℒ₂ is comax complete implies that ℒ₁ is prime complete.

We get Frolík's [8] theorems as our final corollary.

Corollary 6.14. (a) If ℒ₁⊆ℒ₂ and ℒ₂ is ℒ₁-ℵ₀-normal and prime complete, then ℒ₁ is prime complete.
(b) Let 𝑋 be a normal 𝑇₂ space. If 𝑋 is almost realcompact and countably paracompact, then 𝑋 is realcompact.

Acknowledgments

This paper is based on the Ph.D. thesis the author wrote under the supervision of the late George Bachman at The Polytechnic Institute of Brooklyn, now Polytechnic University of NYU. The author is grateful for Professor Bachman’s patient guidance and encouragement. This material was presented at The Second Annual Dr. George Bachman Memorial Conference at St. John's University Manhattan Campus on June 7, 2009. The author appreciates the comments and encouragement provided by Professors Keith Harrow, Pao-Sheng Hsu, and Noson Yanofsky and the questions raised by the anonymous reviewers. This paper is dedicated to the memory of Professor George Bachman.