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Locally Compact Stone Duality

We prove a number of dualities between posets and (pseudo)bases of open sets in locally compact Hausdorff spaces. In particular, we show that (1) Relatively compact basic sublattices are finitely axiomatizable. (2) Relatively compact basic subsemilattices are those omitting certain types. (3) Compact clopen pseudobasic posets are characterized by separativity. We also show how to obtain the tight spectrum of a poset as the Stone space of a generalized Boolean algebra that is universal for tight representations.


Introduction
A number of dualities exist between classes of lattices and topological spaces. Those most relevant to the present paper are summarized below. Unlike the other dualities, the R-lattices in [Shi52] represent sublattices of regular open sets. This means O ∨ N = O ∪ N • rather than O ∪ N . Alternatively, we could consider the lattice elements as representing regular closed sets instead, but then O ∧ N = (O ∩ N ) • rather than O ∩ N . As topological properties are usually expressed in terms of ∪ and ∩ on general open sets, this makes R-lattices somewhat less appealing for doing first order topology. Thus our first goal is to modify the axioms of R-lattices so as to axiomatize basic sublattices of general relatively compact open sets instead. This is the content of §2- §4, summarized in Corollary 4.5, and in §5 we extend this to equivalence of categories in Theorem 5.3, taking appropriate relations as our basic lattice morphisms. A description of this duality seems long overdue, given the classical nature of the above results. Indeed, despite our best efforts to track down the relevant literature, it is entirely possible that we missed a key reference, in which case we apologize for any duplication. Moreover, our axioms in Definition 2.1 still consist of finitely many first order sentences in a language with a single relation ≺. 1 This is why we must still deal with sublattices rather than the entire open set lattice as in [HL78]. Indeed, there cannot be any first order axiomatization of the entire open set lattice of any non-trivial class of Hausdorff topological spaces, as completeness implies any infinite such lattice is uncountable, thus violating the downward Lowenheim-Skolem theorem.
Next we consider basic meet subsemilattices of relatively compact open sets in §6. Here a finite axiomatization is not possible, as explained at the end of §4, however we show that they can still be characterized by omitting types. This is particularly important for one of our original motivations, which will be developed in subsequent papers. Specifically, we wanted to generalize the construction of C*-algebras from inverse semigroups, allowing for combinatorial descriptions of C*-algebras even without real rank zero. For this, one first needs to be able to define an appropriate topological space from the idempotent semilattice and the theory here provides such a construction, given an extra relation ≺ representing 'compact containment'.
Lastly, in §7- §9, we consider pseudobasic p0sets(=posets with minimum 0) of compact clopen sets in the 0-dimensional case, showing that these have a simple finite axiomatization as the separative p0sets. We also use a well-known set theoretic construction to define a generalized Boolean algebra from any p0set that is universal for tight representations. This allows us to identify the tight spectrum of the p0set with the Stone space of the algebra, providing a different take on some of the theory from [Exe08].

Basic Lattices
Assume ≺ is a transitive relation on set B with minimum 0, i.e.
Note that the definition of and the transitivity of ≺ immediately yields x ≺ z y ⇒ x ≺ y. (Left Auxiliarity) x ≺ y ⇒ x y. (Domination) 1 Actually, [Shi52, Definition 2] makes use of two relations ≤ and ≪, as well as a ternary function implicit in part v). However, we believe this could be avoided by a slightly more careful axiomatization in terms of just ≪.
Definition 2.1. We call (B, ≺) a basic lattice if (B, ) is also a lattice and x ≺ y ≺ z ⇒ ∃w ⊥ x (w ∨ y = z).

(Complementation)
As part of the definition of a lattice we take it that is antisymmetric and hence a partial order. Also, by (Coinitiality), (Left Auxiliarity) and (Domination), we could replace ≺ with in the definitions of ⋓ and ⊥, i.e. for basic lattice B x ⋓ y ⇔ x ∧ y = 0.
Note then inclusion ⊆ is indeed the relation defined from ⊂ by (Reflexivization).
Proposition 2.2. If B is a basis of relatively compact open sets that is closed under ∪ and ∩ then (B, ⊂) is a basic lattice.
Proof. We prove the last two properties and leave the rest as an exercise.
In the next section we show that all basic lattices arise in this way from locally compact Hausdorff spaces. Here we just note some more properties of basic lattices. Proposition 2.3. Any basic lattice B satisfies the following. Proof.
(Distributivity) ⇐ is immediate. Conversely, by (Left Auxiliarity) and (Decomposition), for any w ≺ z, we have x ′ ≺ x and y ′ ≺ y with w = x ′ ∨ y ′ . By (Multiplicativity), . (Rather Below) For the ⇒ part, assume x ≺ y. By (Cofinality), we have y ′ ≻ y. By (Left Auxiliarity), y ≺ y ′ ∨ z, for any z ∈ B. By (Complementation), we have w ⊥ x with w ∨ y = y ′ ∨ z z, as required. Conversely, take x, y ∈ B satisfying the right hand side. By (Cofinality), we have z ≻ x and then we can take w ⊥ x with z w ∨ y. By (Left Auxiliarity), x ≺ w ∨ y. By (Decomposition), we have y ′ ≺ y and When B has a maximum 1, it suffices to take z = 1 in (Rather Below) which is the definition of the rather below relation in [PP12, Ch 5 §5.2] and the well inside relation in [Joh86, III.1.1]. In any case, Proposition 2.3 shows that we could equivalently take as the primitive relation in the definition of a basic lattice and define ≺ from as in (Rather Below). 2 Then the definition of from ≺ at the start would become a defining property of a basic lattice instead. Indeed, most treatments of continuous lattices take as the primary notion and define ≺ as the way-below relation from , but we will soon see that there are good reasons to focus more on ≺.
The basic lattice axioms could also be reformulated in several ways. For example, we could combine (Cofinality) and (Complementation) into We could also replace ⇒ with ⇔ in (Decomposition) to combine it with (Additivity). Or we could replace (Decomposition) and (Interpolation) with (Distributivity) and Or we could even avoid the use of meets in (Multiplicativity), as shown below.
Lastly, let us note that when ≺ is reflexive, i.e. when ≺ coincides with , most of the basic lattice axioms are automatically satisfied. Indeed, for a lattice (B, ) to be a basic lattice it need only satisfy (Decomposition), which is then the same as (Distributivity), and (Complementation) which, as we can take x = y, is saying that B is section complemented in the terminology of [Ste99]. In other words, (B, ) is a basic lattice ⇔ (B, ) is a generalized Boolean algebra.
So while we have no unary complement operation like in a true Boolean algebra, we do have a binary relative complement operation x\y, i.e. satisfying (x\y) ∧ (x ∧ y) = 0 and (x\y) ∨ (x ∧ y) = x.

Filters
Definition 3.1. For any transitive relation ≺ on B, we call U ⊆ B a ≺-filter if Throughout the rest of this section, B is an arbitrary but fixed basic lattice.
For the ⇒ part of (3.1), note any ≺-directed U ⊆ B is ≺-coinitial and, by (Domination), -directed. And any ≺-coinitial ≻-closed U ⊆ B is -closed, as x y ∈ U implies y ≻ z ∈ U , for some z, so x ≻ z ∈ U , by (Left Auxiliarity), and hence x ∈ U .
Conversely, for the ⇐ part of (3.1), note any -closed U ⊆ B is ≻-closed, by (Domination). And any ≺-coinitial -directed U ⊆ B is ≺-directed, as any x, y ∈ U satisfies z x, y, for some z ∈ U , and then w ≺ z, for some w ∈ U , and hence w ≺ x, y, again by (Left Auxiliarity).
Ultrafilters in Boolean algebras can be characterized in a couple of first order ways as the proper prime filters or as the proper filters that intersect every complementary pair. These characterizations generalize to basic lattices as follows.
We call a ≻-filter a ≺-ideal.
Proposition 3.3. For non-empty ≺-filter U B, the following are equivalent.
(1) U is a ≺-ultrafilter. ( Proof. (1)⇒(3) For any proper ≺-filter U ⊆ B, For the reverse inclusion, assume y is not in the set on the left, so we have x ≺ y such that w ⋓ x, for all w ∈ U . Thus y ∈ V B and U ⊆ V for Thus if U is a ≺-ultrafilter then V = U and hence y ∈ U , as required.
(2)⇒(3) Assume (2) and take z ∈ U . By (Rather Below), if x ≺ y / ∈ U then we have w ⊥ x with z w ∨ y and hence w ∨ y ∈ U . If w / ∈ U then, as y / ∈ U and B \ U is a -ideal, w ∨ y / ∈ U , a contradiction. Thus w ∈ U . (3)⇒(2) Assume (3) and take x, y / ∈ U . By (Decomposition), for any z ≺ x ∨ y, we have x ′ ≺ x and y ′ ≺ y such that z = x ′ ∨y ′ . By (3), we have u ′ , v ′ ∈ U with u ′ ⊥ x ′ and v ′ ⊥ y ′ . As U is -directed, we have w ∈ U with w u ′ , v ′ and hence w ⊥ x ′ , y ′ . By (Distributivity), w ⊥ z and hence z / ∈ U . As z was arbitrary, x ∨ y / ∈ U . As x, y / ∈ U were arbitrary, B \ U is a -ideal.

Stone Spaces
Definition 4.1. The Stone SpaceB of (B, ≺) is the set of ≺-ultrafilters in B with the topology generated by It follows straight from the definitions that (O x ) x∈B not only generates the topology ofB, but is actually a basis forB.
Proposition 4.2. If X is a locally compact Hausdorff space and B is a relatively compact basis of X then x → B x is a homeomorphism from X ontoB where and ≺ is the relation ⊂ on B defined in (Compact Containment).
Proof. As X is locally compact and B is a basis, every B x is a ⊂-filter. Also As U is a ⊂-filter, the latter collection of compact sets has the finite interesection property. Thus we have some x ∈ U , i.e. U ⊆ B x so, by maximality, This also means that if we had x ∈ X and a filter U ⊆ B properly containing B x then we would have U = ∅, despite the fact U can be represented as an intersection of compact sets with the finite intersection property as above, a contradiction. So we do indeed Note that B above is not even required to be a sublattice of open sets. Thus it is not clear that ⊂ (equivalently the topology of X) can be recovered from ⊆.
Question 4.3. Could the topology of X above be recovered from (B, ⊆) instead?
More interesting is the fact that we have the following converse of Proposition 2.2. Thus we have a duality between basic lattices and locally compact Hausdorff spaces. Proof.
(4.2) Every ≺-filter is a -filter, by Proposition 3.2. (4.3) Again, this follows from Proposition 3.2 and Proposition 3.3 (2). (4.4) This will follow from (4.2), once we show that O x = ∅ ⇔ x = 0. For this, note that if x = 0 then we have y = 0 with y ≺ x, by (Coinitiality). Then {z ∈ B : y ≺ z} is a ≺-filter, by (Riesz Interpolation), which extends to a ≺-ultrafilter U ∈ O x . Now say we have C ⊆ B and z ∈ B with z ≺ F , for all finite F ⊆ C, and let We claim that then If z ≺ x∨ F, y∨G then, by (Multiplicativity), (Distributivity) and (Left Auxiliarity), z ≺ (x∧y)∨ (F ∪G), so D is a -filter. By (∨-Interpolation), D is ≺-coinitial and hence a ≺-filter. As z ≺ F , for all finite (4.5) The claim with C = {y} yields the ⇒ part. Conversely, if x ≺ y and U ∈ O x then x ⋓ z, for all z ∈ U , and hence y ∈ U , by Proposition 3.3 (3). (4.6) The claim with C = ∅ yields ⊇ while (4.5) yields ⊆.
By (4.5), the claim is saying that O x is compact, for all x ∈ B, and henceB is locally compact. To see thatB is Hausdorff, take U, V ∈B. If we had V ⊆ U then we would have V = U , by maximality. So if U and V are distinct then we have Corollary 4.5. Basic lattices characterize ∩-closed ∪-closed relatively compact bases of (necessarily locally compact) Hausdorff spaces. More precisely: We could further replace (Decomposition) with (∨-Interpolation), but this is too weak. Indeed, note that every finite basic join semilattice is isomorphic to the lattice of all subsets P(X) of some finite X (where ≺ coincides with ), and there are plenty of other finite lattices satisfying these axioms, e.g. the diamond lattice D 3 with minimum 0, maximum 1 and three incomparable elements in between. In fact, when ≺ is reflexive and hence coincides with , we can always just take x ′ = x and y ′ = y in (∨-Interpolation).
On the other hand, we know there is no finite axiomatization of basic meet semilattices (defined like basic join semilattices but with ∩ replacing ∪). In fact, basic meet semilattices do not even form an elementary class. To see this note that, for every n ∈ N, D n is a basic meet semilattice (representing a basis of the 5 By [Wol56, Theorem 2], the basic join semilattices with a maximum (corresponding to the compact case) are precisely those join semilattices satisfying the dual of separtivity in which every maximal Frink ideal is prime. However, the mention of subsets here, namely ideals, makes this characterization second order rather than first order. discrete space with n points), but an ultraproduct of (D n ) is not. Indeed, such an ultraproduct is isomorphic to D κ for some (uncountable) infinite κ. If such a lattice represented a basis of relatively open sets closed under ∩ in locally compact Hausdorff X, each element of D κ \ {0, 1} would represent an isolated point, while 1 would represent X which, being relatively compact by definition, must actually be compact. As infinite collections of isolated points are not compact we must have some other point x ∈ X. As X is Hausdorff and B is a basis, there must be some proper open set containing x represented in D κ , a contradiction. We will return to this problem in §6.
Lastly, let us note that ⊂ is reflexive precisely when the basis elements are not just open but also compact. Thus we get a duality between 0-dimensional locally compact Hausdorff spaces and generalized Boolean algebras. And a ∪-closed basis has a maximum precisely when X is compact, so in this case we recover the classical Stone duality between 0-dimensional compact Hausdorff spaces and Boolean algebras.

Interpolators
If we really hope to do topology in a first order way, we also need a first order analog of continuous maps. For this, we introduce interpolators. Actually, (Interpolation) for ⊏ above really becomes two axioms Proposition 5.2. If ⊏ is an interpolator then we also have Proof. If x ⊏ z ≤ y then 0 ⊏ y, by (Minimum), so x = 0 ∨ x ⊏ y ∨ z = y, by (Additivity). Thus if x z ⊏ y then we have w ⊐ x, by (Cofinality), so x = x ∧ z ⊏ w ∧ y y, by (Multiplicativity), and hence x ⊏ y, by (≤-Auxiliarity).
We define composition of relations in the usual way, namely by It is routine to verify that a composition of interpolators is again an interpolator. It is also immediate from the definitions that if (B, ≺) is a basic lattice then ≺ is an interpolator from B to itself with ≺ = ≺ • ≺, as ≺ is transitive and satisfies (Interpolation). So taking interpolators as morphisms turns the class of basic lattices into a category, which we denote by BasLat. We also let LocHaus denote the category of locally compact Hausdorff topological spaces with continuous maps as morphisms.
If x ⊏ y then (≺-Interpolation) yields z ∈ B with x ≺ z ⊏ y. Then (4.5) yields Conversely, assume x ⊏ y and let By ≺-Auxiliarity, if 0 ∈ D then x ⊏ y, a contradiction, so 0 / ∈ D. Again using ≺-Auxiliarity and arguing as in the proof of Theorem 4.4, D can be extended to On the other hand, if f : X → Y is a continuous map between locally compact Hausdorff spaces with relatively compact bases B and C respectively, it is routine to verify that we get an interpolator ⊏ defined by Then we immediately see that C f (x) = B ⊏ x , for all x ∈ X.

Basic Semilattices
Our next goal is to characterize basic meet semilattices by omitting types. By a 'type' we mean a collection of first order formulas and by 'omit' we mean that there are no elements which satisfy the entire type (see [Mar02,Ch 4]). Specifically, consider the types (φ n ) and (ψ n ) where φ n (x, y) ⇔ x ≺ y and ∀w 1 , . . . , w n ≺ y ∀v 1 ≺ w 1 . . . ∀v n ≺ w n (φ n ) ψ n (x, y, z) ⇔ x ≺ y and ∀y ′ ≺ y ∀w 1 , . . . , w n ⊥ x ∀v 1 ≺ w 1 . . . ∀v n ≺ w n (ψ n ) As we shall soon see, omitting (φ n ) corresponds to (Interpolation), while omitting (ψ n ) corresponds to (≺-Below). Also let (θ n ) be the sentences given by Consider the relations defined on P(B) by is reflexive too. Also define (6.1) C ∧ D = {x ∧ y : x ∈ C and y ∈ D}.
too. ( -additivity) As ( P) ≻ = A∈P A ≻ and ( P) ⋓ = A∈P A ⋓ , for any P ⊆ P(B), the -additivity of ≺ and follows immediately from the -additivity of ⊆.
The converse is immediate from C ≻ ⊆ C .
If C ≺ ≺ D then we have F ∈ F (B) with C F ≺ D so, as ≺ is stronger than , C F D and hence C D, again by the transitivity of .
Proposition 6.4. For all A, C ⊆ B and F ∈ F (B), Proof.
Thus ∪-additivity yields F y∈F G y ≺ A and hence F ≺ ≺ A.  by (6.4). Conversely In the next result we use some standard terminology from frame and domain theory (see e.g. [GHK + 03], [PP12] or [GL13]). Specifically, by a frame we mean a complete lattice L where finite meets distribute over arbitrary joins. For x, y ∈ L, we say x is way-below y if x ∈ Z whenever y Z for -directed Z. And we say L is continuous if every x ∈ L is the join of those elements way-below x.
(6.8) For all A ∈ P, A ⊆ P so A ∪ ⊆ ( P) ∪ . Conversely, if O ∈ P(B) ∪ and A ∪ ⊆ O, for all A ∈ P, then, by Proposition 6.4, (Distributivity) Take A ⊆ B and P ⊆ P(B). By (6.9), (Way-Below) Take C ⊆ B so C ∪ = C ≻∪ = F ∈F (C ≻ ) F ∪ and this latter union is directed. Thus if D ⊆ B and D ∪ is way-below C ∪ in P(B) ∪ then by definition By (6.2) and (6.6), F ∪ ≺ ≺ C ∪ , for all F ∈ F (C ≻ ), so the continuity of P(B) ∪ will follow from the converse. For this, assume D ≺ ≺ C ⊆ A∈P A ∪ , for some C, D ⊆ B and P ⊆ P(B). So we have F ∈ F (B) with As this last union is directed, we have G ∈ F ( P) with D F ≺ G ∪ , i.e. D ≺ ≺ G ∪ so D ≺ ≺ G, by (6.6), and hence D ⊆ G ∪ . Taking finite G ⊆ P with G ⊆ G yields D ⊆ ( G) ∪ = A∈G A ∪ . As P was arbitrary, this shows that D is way-below C in P(B) ∪ , as long as C and D are in P(B) ∪ .
Recall that we denote the singleton subsets of B by S(B) = {{b} : b ∈ B}. We call a subset S of a lattice L -dense (∨-dense) if every element of L is a (finite) join of elements in S. Corollary 6.6.
(Distributivity) By Theorem 6.5, (P(B) ∪ , ⊆) is distributive, thus so is any sublattice. (∨-Interpolation) This holds for any -dense sublattice of a continuous lattice, so again this follows from Theorem 6.5. (Additivity) Likewise, this holds for any join subsemilattice of a continuous lattice.
(≺-Below) As B omits (ψ n ), for any x, y, z ∈ B with x ≺ y, we have y ′ ≺ y and By Proposition 6.2, U ≺ Y , V ≺ W ⊥ F X and Z U ∪V ≺ Y ∪W so W ⊥ X and Z ≺ ≺ Y ∪W . Thus F (B) ∪ satisfies (≺-Below), by (6.6) and (6.8).
Proposition 6.7. If X is a locally compact Hausdorff space with ∩-closed basis B of relatively compact clopen sets then (B, ⊂) is a basic semilattice, Proof.
As F is compact, C is also compact so C ⊂ D. Conversely, if C ⊂ D then, for each As O is compact, we have some subcover of size n < ∞, showing that φ n (O, N ) fails. Similar compactness arguments show that B omits (ψ n ) and satisfies (θ n ). Also (Coinitiality) and (Multiplicativity) are immediate so B is a basic semilattice.
By (6.11), P → P is an isomorphism from (P (B Corollary 6.8. Basic semilattices characterize ∩-closed relatively compact bases of (necessarily locally compact) Hausdorff spaces. More precisely, if B is a relatively compact basis of Hausdorff X then (B, ⊂) is a basic semilattice andB is homeomorphic to X, while if (B, ≺) is a basic semilattice thenB has a relatively compact basis isomorphic to B.
Proof. If B is a relatively compact basis of Hausdorff X then (B, ⊂) is a basic semilattice, by Proposition 6.7, andB is homeomorphic to X, by Proposition 4.2.
If (B, ≺) is a basic semilattice then B is isomorphic to a ∨-dense meet subsemilattice of the basic lattice (F (B) ∪ , ⊂), by Corollary 6.6. Thus B is isomorphic to a ∩-closed relatively compact basis of a (locally compact) Hausdorff space, by Corollary 4.5. By Proposition 4.2, this space is homeomorphic toB.
In other words, the basic semilattices of this section are the same as the basic meet semilattices defined at the end of §4. Part of the above theorem could also be obtained from the Hofmann-Lawson theorem from [HL78]. Specifically, as B is isomorphic to a -dense meet subsemilattice of the continuous frame (P(B) ∪ , ⊂), by Corollary 6.6, B must be isomorphic to a ∩-closed basis of a locally compact sober space.
As with basic lattices, the basic semilattice axioms can be much simplified when ≺ is reflexive. Specifically, for a meet semilattice (B, ) to be a basic semilattice, it suffices to omit (ψ n ) and satisfy θ 1 , which becomes While this is still not a finite axiomatization, we show in §9 that more general 'pseudobases' of compact clopen sets can be axiomatized by (Separativity) alone.

Tight Representations
We call a poset (B, ) with minimum 0 a p0set and apply all our previous notation and terminology to p0sets by taking ≺ = . For C ⊆ B, let We take the empty intersection to be the entire p0set, i.e. ∅ = B.
Definition 7.1. For C, D ⊆ B we define the covering relation by Unlike the other relations we have been considering, need not be transitive. However, is at least reflexive on P(B)\{∅}. Also, can often be expressed in more familiar order theoretic terms, e.g.
Also note the following relationships between and .
Example 7.3. Let B = {0, x, y} be the meet semilattice with x ∧ y = 0. So ∅ {x, y} is the only non-trivial covering relation. Thus any β : B → A with β(0) = 0 is tightish, while β is a tight representation iff A is a Boolean algebra with maximum β(x) ∨ β(y). For example, β is not tight when we define β : B → P({1, 2, 3}) by However, note that if we restrict the codomain to P({1, 2}) then β is tight.
In general, we see that tightish and coinitial ⇒ tight ⇒ tightish, and they all coincide if we restrict the codomain to β [B]. Also if B 1 denotes B with maximum 1 adjoined and β 1 denotes the extension of β to B 1 with β(1) = 1(∈ A 1 ), If there is no G ∈ F (B) with ∅ G then tight and tightish again coincide. Even when we do have G ∈ F (B) with ∅ G, to verify that tightish β : B → A is tight we only need to check that ∅ β[G] for some (rather than all) such G.
Proof. For any C, D ⊆ B, Thus any tightish β also preserves on Here, 'tight' generalizes [Exe08, Definition 11.6] (and Proposition 7.4 generalizes [Exe08, Lemma 11.7]) while 'tightish' generalizes 'cover-to-join' from [DM14]. The original definitions were restricted to representations of meet semilattice B, in which case we have the following alternative description.
Proposition 7.5. A representation β of a meet semilattice B is tightish iff for all x, y ∈ B and G ∈ F (B). Also, β is tight iff moreover, for all G ∈ F (B), Proof. Assume β is tightish. Then β is order preserving because A is separative so Thus β(x ∧ y) β(x) ∧ β(y). Conversely, by the definition of meets, we have . On the other hand, if (7.2) holds and F G, for F, G ∈ F (B), then we have In particular, if there is G ∈ F (B) with ∅ G then all tight representations of B must be to true Boolean algebras, as in [Exe08, Definition 11.6]. Actually, if we were being faithful to [Exe08, Definition 11.6], we would define B C,D = C ∩ D ⊥ = {e ∈ B : ∀x ∈ C(x e) and ∀y ∈ D(y ⊥ e)} and call β tight if β(0) = 0 and, for all F, G, H ∈ F (B), However, this is equivalent to our definition as If we restrict further to generalized Boolean algebra B, we see that the tightish representations are precisely the generalized Boolean homomorphisms, i.e. the maps preserving ∧, ∨ and \. Indeed, we will soon see how the category of posets with tightish morphisms is in some sense a pullback of the category of generalized Boolean algebras with generalized Boolean morphisms.
Proposition 7.6. For generalized Boolean algebras A and B and β : B → A, the following are equivalent.
Proof. By the observations after Definition 7.1, in any generalized Boolean algebra, Thus, arguing as in the proof of Proposition 7.5, we see that the tight maps between generalized Boolean algebras are precisely the lattice homomorphisms taking 0 to 0. As x\x = 0 and x\y is the unique complement of x ∧ y in [0, x], these are precisely the generalized Boolean homomorphisms.
Proposition 7.7. For Boolean algebras A and B and β : B → A, the following are equivalent.

The Enveloping Boolean Algebra
Next we construct a tight map from any given p0set B to what might be called its 'enveloping Boolean algebra' RO(B ′ ). We then examine its universal properties.
First, let B ′ = B\{0} with the Alexandroff topology, where the closed sets are precisely the -closed sets, and consider the map x → {x} \{0} from B to O(B ′ ).
For the reverse inclusion, it suffices to show that O If this inclusion failed, we would have Proof. For any Y ⊆ B ′ , we see that Thus Thus a representation β of a p0set B is tight iff, for F, G ∈ F (B).
We now show that ρ restricted to the generalized Boolean subalgebra of RO(B ′ ) generated by ρ[B] is universal for tight(ish) representations.
Theorem 8.4. Let β : B → A be a representation of a p0set B, let ρ be as in (8.1), and let S be the generalized Boolean subalgebra of RO(B ′ ) generated by ρ [B]. Then β is a tight(ish) representation iff β factors through ρ, i.e. iff there is tight(ish) π from S to A such that β = π • ρ.
We can then extend π to the meet semilattice M generated by ρ[B] by defining . It the follows from the defintion that this extension to M is meet preserving.
As RO(B ′ ) is distributive, the lattice L generated by ρ[B] is generated by joins of elements of M . We claim we can extend π to L by defining, for F ∈ F (M ), As w = (w ∧ x) ∨ (w\x), for all w ∈ S, L ⊆ L x and it suffices to take y ⊆ x above. Then we claim that any lattice homomorphism π from L to A can be extended to a lattice homomorphism π ′ of L x given by π ′ (y ∨ (z\x)) = π(y) ∨ (π(z)\π(x)).
So π ′ extends π and likewise π ′ is verified to be a lattice homomorphism.
Thus any maximal lattice homomorphism extension of π defined on the sublattice generated by ρ[B] as above must in fact be defined on the entirety of S. Thus π is tightish, by Proposition 7.6. If there is no G ∈ F (B) with ∅ G then π is even (vacuously) tight. While if G ∈ F (B), ∅ G and β is tight then ∅ β[G] = π •ρ[G] so π is also tight, by Proposition 7.4.
It follows that tight(ish) maps between general p0sets are precisely those coming from tight(ish) maps of generalized Boolean algebras.
Corollary 8.5. For p0sets A and B with ρ A : A → S A and ρ B : B → S B as above, Proof. If β is tight(ish) then so is ρ A •β and the required π comes from Theorem 8.4. On the other hand, if ρ A • β = π • ρ B and π is tight(ish) then so is π • ρ B and hence ρ A • β. This means, for all F, G ∈ F (B) (with F = ∅), by Proposition 8.3, so β is tight(ish) too.
Thus we have a map β → π β taking any tight(ish) β : B → A to the unique tight(ish) π β : S B → S A satisfying ρ A • β = π β • ρ B . To put this in category theory terms, let P denote the category of p0sets with tight(ish) morhpisms and let G denote its full subcategory of generalized Boolean algebras. The above results are saying that we have a full functor F from P onto G with F (B) = S B and F (β) = π β together with a natural transformation ρ from the identity functor I to F : . The Tight Spectrum Definition 9.1. Let B be any p0set. The tight spectrumB is the space of non-zero tight characters on B taken as a subspace of {0, 1} B with the product topology.
We could equivalently call this the tightish spectrum, as any non-zero tightish character is coinitial and hence tight. And if ∅ G, for some G ∈ F (B), then every tight character is automatically non-zero and so our definition ofB agrees with the definition ofB tight from [Exe08, Definition 12.8]. When there is no G ∈ F (B) with ∅ G, we instead haveB tight =B 1 = the one-point compactification ofB, where B 1 here denotes B with a top element 1 adjoined.
We can also view the tight spectrum as a certain Stone space, for by Theorem 8.4, we can identifyŠ andB via the map φ → φ • ρ. We can then identifyŠ withŜ via the map φ → φ −1 {1}, as non-zero tight characters on generalized Boolean algebras are precisely the characteristic functions of ultrafilters (as lattice homomorphisms from generalized Boolean algebras to {0, 1} are precisely the characteristic functions of prime filters).
Definition 9.2. For any topological space X, we call B ⊆ O(X) a pseudobasis if . Also letḂ denote the characteristic functions of maximal centred C ⊆ B, i.e. satisfying F = {0}, for all F ∈ F (C).
Proof. As {0, 1} B is 0-dimensional Hausdorff, so isB. The tight characters are immediately seen to form a closed subset of {0, 1} B so taking away the zero character still yields a locally compact spaceB.
As O 0 = ∅, (O x ) x∈B satisfies (Minimum). As every φ ∈B has value 1 for some x ∈ X, (O x ) x∈B satisfies (Cover). If φ, ψ ∈B are distinct then φ(x) = ψ(x), for some  Theorem 9.4. If B is a pseudobasis of compact clopen subsets of a topological space X then we have a homeomorphism from X ontoB given by Proof. For any F, G ∈ F (B), we claim that If Thus φ x is tight and also non-zero, by (Cover), so φ x ∈B. We next claim that, for any φ ∈B, there is a unique {x} such that O\N. On other other hand, the intersection can not contain more than one point, by (T 0 ). This proves the claim, which means φ = φ x . Thus x → φ x is a bijection from X toB. Now say we have x ∈ M ∈ O(X). By (T 0 ),

O\N.
As each O ∈ B is compact clopen and B satisfies (Cover), some finite subset has empty intersection, i.e. we have F, G ∈ F (B) with x ∈ F , x / ∈ G and O∈F,N ∈G O\N ⊆ M . As x and M were arbitrary, this is saying x → φ x is an open mapping. As each O ∈ B is clopen, x → φ x is also continuous and hence a homeomorphism.
Corollary 9.5. Separative p0sets characterize compact clopen pseudobases of necessarily 0-dimensional locally compact Hausdorff topological spaces. More precisely: If B is a compact clopen pseudobasis of X then (B, ⊆) is separative andB is homeomorphic to X, while if (B, ) is separative thenB has compact clopen pseudobasis (O x ) x∈B order isomorphic to B.
Proof. If B is a compact clopen pseudobasis of X then, for any O, N ∈ B with O N , we see that ∅ = O\N ∈ O(X). By (Coinitiality), we then have non-empty M ∈ B with M ⊆ O\N , so B is separative. By Proposition 9.3, X is homeomorphic toB, which is 0-dimensional locally compact Hausdorff, by Proposition 9.3.
If (B, ) is separative then, whenever x y, we have non-zero z x with y ⊥ z. We can then take φ ∈Ḃ with φ(z) = 1 so φ ∈ O x \O y and hence O x O y . Conversely, if x y then O x ⊆ O y so We finish with a note on (Separativity), which is the standard term in set theory (see [Kun80, Ch II Exercise (15)]), and some other closely related conditions. First note that (9.1) implies that x → O x is injective, as O x = O y then implies x y x so x = y. This is equivalent to saying that ρ from (8.1) is injective. This, in turn, implies that B is 'section semicomplemented' in the sense of [MM70,Definition 4.17], specifically (SSC) x = y x ⇒ ∃z = 0 (y ⊥ z x).