Adjoint maps between implicative semilattices and continuity of localic maps

We study residuated homomorphisms (r-morphisms) and their adjoints, the so-called localizations (or l-morphisms), between implicative semilattices, because these objects may be characterized as semilattices whose unary meet operations have adjoints. Since left resp. right adjoint maps are the residuated resp. residual maps (having the property that preimages of principal downsets resp. upsets are again such), one may not only regard the l-morphisms as abstract continuous maps in a pointfree framework (as familiar in the complete case), but also characterize them by concrete closure-theoretical continuity properties. These concepts apply to locales (frames, complete Heyting lattices) and provide generalizations of continuous and open maps between spaces to an algebraic (not necessarily complete) pointfree setting.


Introduction
A basic tool in order theory with countless applications in other fields of mathematics is provided by adjoint pairs of maps between partially ordered sets (posets): given posets A, B and maps h : A −→ B and f : B −→ A related by the equivalence f is the (right or upper) adjoint of h, and h the coadjoint (left or lower adjoint) of f (we omit parentheses if maps are applied to elements). Either partner of an adjunction is uniquely determined by the other. The letter h has been chosen because in our investigations h often will be an algebraic homomorphism, whereas f will represent certain continuous functions. In Section 5, g will stand for the left (French: gauche) adjoint of h, provided it exists. In other contexts, it is more common to denote a left adjoint by f and its (right) adjoint by g; accordingly, in the categorical version of adjointness [1,22], left adjoint functors generalize free functors , and their adjoints grounding functors . Some authors use the opposite order ≥, so that an upper adjoint g stands on the left side of the inequality gb ≥ a [19].
The adjoining equivalence allows to shift parts of one side of an inequality to the other side in a very convenient way. It is well known and easy to see that a map is coadjoint (left adjoint) iff it is residuated, i.e., preimages of principal downsets are again principal downsets, and a map is (right) adjoint iff it is residual, i.e., preimages of principal upsets ↑c = {a ∈ A | c ≤ a} are again principal upsets. Residuated maps preserve all existing joins, and residual maps all existing meets. In particular, both kinds of maps are isotone (order-preserving). Moreover, a map between complete lattices is residuated (resp. residual) iff it preserves arbitrary joins (resp. meets). The most important fact in the theory of (Galois) adjunctions is that any adjoint pair of maps induces mutually inverse isomorphisms between their ranges. Note also that one partner of an adjunction is idempotent iff so is the other, that it is injective iff the partner is surjective, and that passing to adjoints inverts composition; see [8,13,19].
A basic instance of an adjunction is obtained as follows. Every map f : S −→ T between sets induces an adjoint pair of maps between the power sets PS and PT : the image map They are related by the equivalence Topologies are prototypes of so-called frames or locales [22,37], that is, complete lattices in which binary meets distribute over arbitrary joins. The category Frm of frames has as morphisms the frame homomorphisms, that is, maps which preserve arbitrary joins and finite meets, while the opposite category Loc of locales has the same objects but as morphisms the adjoints of frame homomorphisms, so-called locale morphisms or localic maps. Denoting for any topological space T its topology, regarded as a frame resp. locale, by OT , we obtain for every continuous map f : S −→ T between topological spaces This may be regarded as the "starting point of pointfree topology" (cf. Johnstone [23]). Comprehensive references to themes of pointfree topology are Dowker and Papert [10], Isbell [21], Johnstone [22,23,24], Picado, Pultr and Tozzi [36,37,39], and Simmons [41,42,43].
For the important special case where S is a subspace of a space T and e is the inclusion map from S into T , the above construction yields an adjoint pair of maps which is a meet-closed and left →-closed subset, that is, a sublocale of OT [36,37]. In this sense, one may say that sublocales represent subspaces; however, the map O T from PT to the coframe of all sublocales is neither one-to-one nor onto in general. For topological characterizations of those spaces for which O T is injective, surjective or bijective, respectively, see [37] and [41].
In view of the connections between spaces and locales, categorically inspired authors refer to localic maps (or to the opposite arrows of frame homomorphisms [22]) as "continuous maps". This raises the question of whether that kind of maps may be characterized by certain concrete continuity properties in the closure-theoretical sense (preimages of closed subobjects are closed, and the formation of preimages commutes with complementation). We shall give an affirmative answer to that question; the explicit characterization of localic maps in terms related to continuity is, however, a bit delicate: the complements of closed sublocales have to be formed in the lattice of all sublocales and not set-theoretically as in classical topology.  Motivated by the previous observations, we shall study, more generally, implicative semilattices, that is, meet-semilattices with top elements in which the unary meet operations λ a = a ∧ − have adjoints α a = a → −. The frames resp. locales are just the complete implicative semilattices, and the frame homomorphisms are nothing but the residuated semilattice homomorphisms preserving the top elements. Our arguments are often shorter than those found in the literature for the case where joins exist, and nevertheless provide proper extensions to the setting of semilattices; new ideas are required when certain joins or meets are not available.
If we wish to have Frm resp. Loc as full subcategories of two respective dual categories whose objects are implicative semilattices, we have to consider as morphisms not the usual implicative homomorphisms (which preserve finite meets and the binary residuation →) but the residuated top-preserving semilattice homomorphisms, briefly referred to as r-morphisms, and in the opposite direction their adjoints, the so-called localizations (Bezhanishvili and Ghilardi [6]) or l-morphisms. Thus, r-morphisms have right adjoints and preserve finite meets, l-morphisms have left adjoints that preserve finite meets, and the respective categories are duals of each other via Galois adjunction.
For continuous maps f between spaces, O ← f is an r-morphism and O → f its adjoint l-morphism.
Under the point of view we adopt in the present paper, it is reasonable to regard implicative semilattices as algebras (A, ∧, , α), where α is the family of all unary residuations or relative pseudocomplementations α a (a ∈ A). Here we leave the classical area of varieties, because the signature depends on A, and the subalgebras are those subsemilattices which are closed under each α a ; we call them l-ideals. Now, all unary meet operations become r-morphisms, and the image (but not the preimage) of an l-ideal under an l-morphism is always an l-ideal. Those subsets for which the inclusion map is an r-resp. l-morphism will be referred to as r-resp. l-domains. In the complete case of frames/locales, the r-domains are the subframes, whereas the l-domains are the sublocales. In our general setting, they are still nothing but the ranges of nuclei, that is, closure operations preserving finite meets.
The idea to characterize algebraic homomorphisms and their categorical duals by continuity properties is central in the development of general Stone duality [14], and also in the present context, morphisms receive a concrete topological flavor: regarding principal upsets as basic closed sets renders adjoint maps "basic continuous": preimages of basic closed sets are basic closed. More precisely, we justify the term "continuous" for locale morphisms by showing that the l-morphisms between implicative semilattices are characterized by the following continuity condition: the preimage of the zero ideal (the least basic closed set) is zero, the preimage of any basic closed set is basic closed, and its complement in the lattice of l-ideals is contained in the preimage of the complement-a triviality in the counterpart of set-theoretical complements, but an unavoidable additional condition in the lattice-theoretical setting. Vol. 83 (2022) Adjoint maps between implicative semilattices Page 5 of 23 13 In the complete case of locales, the prefix "basic" is omitted, since the basic closed sets then form a closure system, that is, a collection of sets closed under arbitrary intersections (with ∅ being the entire ground set). The closed sublocales and their lattice-theoretical complements, the open sublocales, represent (via O T ) closed resp. open subspaces. In a suitable categorical framework, they also correspond to Isbell's abstract open resp. closed parts [21]. If for an l-morphism f : B −→ A and an l-domain C of A there is a greatest ldomain D of B contained in f ← C then D is called the localic preimage of C. Such localic preimages exist in the complete case of locales, but not in general. Four questions arise immediately: (1) Is any adjoint map whose preimages of opens are open an l-morphism?  [26] about open localic maps to the non-complete situation, by establishing a dual isomorphism between the category of implicative semilattices with basic open lmorphisms as morphisms and the category having the same objects but as morphisms those implicative maps which are biadjoint, that is, both adjoint and coadjoint; in fact, these are just the coadjoints of the basic open l-morphisms. More generally, via Galois adjunction, arbitrary biadjoint maps correspond to the quasi-open l-morphisms. Similar phenomena have been observed in other contexts (cf. Erné [12], Hofmann and Mislove [20]).
Finally, in order to bring together all pieces of the puzzle, we introduce a category of basic zero-dimensional spaces, similar to categories considered in [11] and [14]. The objects are closure spaces with a distributive closure system containing a specified meet-base of complemented members. All the categories discussed here, and many more, like that of T D -spaces (Aull and Thron [2]), front spaces (Skula [44]), and of course, Stone spaces and the more general zero-dimensional spaces (Johnstone [22]), are embedded in the category of basic zero-dimensional spaces. Hence, that category might be an interesting subject of future research.

Closure operations, closure ranges and adjunctions
Let A be a partially ordered set (poset), ≤ its order relation, and A op the dual poset. We write b = a ∨ c if b is the least upper bound (supremum, join), and dually b = a ∧ c if b is the greatest lower bound (infimum, meet) of {a, c}. This convention also applies when not all binary suprema resp. infima exist (that is, not only in lattices but in arbitrary posets). Recall that completeness is a self-dual property: all subsets have joins iff all subsets have meets. A least element (bottom) of A is denoted by ⊥ or ⊥A, and a greatest element (top) by Any set of operations on A is ordered pointwise. A closure operation or hull operation is an isotone (order-preserving), inflationary (extensive) and idempotent operation. The dual notion is coclosure or kernel operation. Closure operations j may be characterized by the single equivalence Other names for such subsets are closure system, partial ordinal [3], or relatively meet-closed set [18,45]. Indeed, any closure range is closed under all existing meets, and the closure ranges of a complete lattice are exactly its meet-closed subsets. We reserve the term "closure system" for sets that are closed under intersections and consequently complete lattices with respect to the inclusion order. Recall that a closure system is topological if it is closed under finite unions, and algebraic if it is closed under directed unions. The term "closure range" is justified by the following fact [3,13,33]: Proposition 2.1. Sending each closure operation to its range, one obtains a dual isomorphism between the pointwise ordered set of all closure operations on A and the set of all closure ranges in A, ordered by inclusion.
Note that h is a closure operation iff h 0 is adjoint to h 0 , and h is a homomorphism iff h 0 and h 0 are homomorphisms. For easy reference, we record the main connections between closure operations and adjoint maps (Blyth and Janowitz [8], Erné [13,15]). For any closure operation j on B with range C, the "image" fjh is a closure operation on A with range fC. Hence, residual maps send closure ranges to closure ranges.

Morphisms between implicative semilattices
By a semilattice we always mean a ∧-semilattice with top element. As morphisms between semilattices we take residuated maps that preserve finite meets, called r-morphisms ; their adjoints are referred to as localizations or l-morphisms. Note that if the codomain of an injective r-or l-morphism is complete then so is the domain, and in the opposite direction, if the domain of a surjective r-or l-morphism is complete then so is the codomain.
An implicative semilattice [6,7,34] (or Brouwerian semilattice [27,28]) is a semilattice whose unary operations λ a = a ∧ − have adjoints α a = a → −: As announced in the introduction, we regard implicative semilattices as algebras (A, ∧, , α) with the family α = (α a : a ∈ A) of unary residuations. Recall that each of the corestricted maps λ a : A −→ ↓a is an r-morphism. Observe that an r-morphism between implicative semilattices preserves not only joins but also complements, to the extent they exist.
Deviating from [40], we reserve the terms Heyting semilattice and Heyting lattice for bounded implicative semilattices resp. lattices (having a least element ⊥). In the lattice case, (A, ∨, ∧, , ⊥, →) is a Heyting algebra. All these algebraic structures are equationally definable (see, e.g., Esakia [18] or Köhler [27,28]). In Heyting semilattices, the element ¬ a = a → ⊥, also denoted by a ⊥ or a * , is the pseudocomplement or negation of a. From now on, A denotes an implicative semilattice with top element . By an interior operation we mean a kernel operation preserving finite meets. On the other hand, a nucleus (see, for example, Bezhanishvili and Ghilardi [6], and for the complete case, Banaschewski [4], Johnstone [22], Simmons [43]) is a closure operation j preserving finite meets; instead of the latter condition, it suffices to postulate the seemingly weaker but equivalent inequality There is a description of nuclei on implicative semilattices by one equation, due to Macnab [31], who calls nuclei on Heyting algebras modal operators: Notice that every nucleus j fulfils the inequality j(x → y) ≤ jx → jy but equality need not hold, that is, j need not be implicative (preserve the formal implication →). An inner characterization of the ranges of nuclei is provided by the next definition: a nuclear range [16] (modal subalgebra in [31], strong ideal in [40]) is a closure range C that is left →-closed, or l-closed, 13  i.e. closed under the unary operations α a , which means that a → c ∈ C for all a ∈ A and c ∈ C. As a closure range contains all existing meets of subsets, every nuclear range is an l-ideal, that is, an l-closed subsemilattice (total subalgebra in [28], ideal in [40]). By definition, the l-ideals are left ideals with respect to the operation →; all order-theoretical filters (dual ideals), i.e. nonempty ∧-closed upsets, are l-ideals, but not conversely.
We denote by T A the algebraic closure system of all l-ideals (total subalgebras), by SlA its ∨-subsemilattice of those l-ideals that are closure ranges, and by N A the same set, but ordered by dual inclusion. Notice that SlA need not be a closure system if A is not complete. The zero ideal 0 = { } is the least element of SlA but the greatest element of N A. The subsequent description of the members of SlA resp. N A is familiar in the more restricted theory of frames and locales, where they are known as sublocales [22,37]. The case of Heyting algebras, due to Macnab [5,30,31], extends without any alteration to implicative semilattices. Proposition 3.1. Sending each nucleus to its range yields an isomorphism between the semilattice NA of all nuclei and the semilattice NA of all nuclear ranges. Hence, these are not only the ranges of nuclei but also the l-domains, that is, those subsets for which the inclusion map into A is an l-morphism.
Analogously, by an r-domain we mean a subset for which the inclusion map is an r-morphism. In light of our general remarks on adjunctions in Section 2, we draw the following conclusions:   In the complete case of frames resp. locales, the r-domains are just the subframes, and on the other hand, the l-domains are just the sublocales. Categorically thinking people mean by a sublocale an extremal l-monomorphism between locales or an extremal r-epimorphism between frames [21,22,39]. In view of Corollary 3.3 all three interpretations are well compatible.
Let us recall a few facts concerning T A and SlA (cf. [22,31,37] for the case of frames, where SlA is a closure system). Binary joins in T A and SlA are given by The next proposition from [16] generalizes results in [28] and [40].
holds whenever I is finite or C is a nuclear range. In particular, (1) T A is an algebraic frame, (2) SlA is a coframe whenever it is a closure system. Hence, in the latter case, the isomorphic lattices NA and NA are frames.
For each a ∈ A, the adjoint map α a = a → − is known to be a nucleus (see, e.g., Macnab [31]), and its range is The following results are from [40] (cf. [37] and [39] for the case of locales): There is also a canonical embedding c A = c of A in (T A) op , sending a to the principal upset ca = ↑a, which is always an l-ideal but need not be nuclear unless A is a lattice, in which case γ a = a ∨ − is the associated nucleus. We record a result that is known for Heyting algebras [31] and frames [37,40]; it extends, by a different argument given in [16], replacing a∨x with (a → x) → x, to implicative semilattices. and closed subspaces to closed sublocales. Observe that even for frames A, the lattice-theoretical complements in T A, in SlA and in its dual NA differ from the set-theoretical complements; see [16,37] for details.

l-morphisms as continuous maps
We now turn to a more thorough investigation of r-morphisms and their adjoints, the l-morphisms.

Proposition 4.2. The image of an l-ideal (l-domain) under an l-morphism f is an l-ideal (l-domain). For injective f the preimage of an l-ideal is an l-ideal. For surjective f, a set is basic closed iff its preimage under f is basic closed.
Proof. Let f : B −→ A be an l-morphism with coadjoint h, and let D be an l-ideal of B. The image fD is a subsemilattice of A (as f preserves meets). For a ∈ A and b ∈ D, we get a → fb = f (ha → b) ∈ fD by Proposition 4.1. Thus, fD is an l-ideal. By Propositions 2.3 and 3.5, fD is a closure range resp. l-domain if D is one. If f is injective then h is surjective. For each l-ideal C of A, the preimage f ← C is an l-ideal, being a subsemilattice such that for b, c ∈ B with fb ∈ C there is an a ∈ A with c = ha, hence f (c → b) = f (ha → b) = a → fb ∈ C and c → b ∈ f ← C. Vol. 83 (2022) Adjoint maps between implicative semilattices Page 11 of 23 13 Now, suppose f is surjective and f ← C is basic closed, say f ← C = ↑b. Then, for a = fb we get fha = a ∈ C, hence b ≤ ha, and then Thus, C = ↑a is basic closed.
The following example demonstrates that the preimage of an l-domain under an injective l-morphism need not be an l-domain.
| n ∈ N} has no join, and the nuclear l-ideal D = {− 1 n | n ∈ N} ∪ {1} has neither in SlA nor in NA a pseudocomplement [16]. Define maps f and g on A by . f and g are l-morphisms with range C, but the preimage of the l-domain B is in both cases the filter F = { 1 n | n ∈ N}, which is not an l-domain. While f is injective but not a nucleus, g is not injective but a nucleus.
In analogy to semilinear maps between vector spaces, modules and algebras, we call a map f : B −→ A semilinear with respect to binary operations on A and B, both denoted by * , if it has a coadjoint h : A −→ B satisfying the Frobenius identity (cf. [9,29,32,36]) In that case, we also say f is * -semilinear. From Proposition 4.1, we deduce one algebraic and one closure-theoretical characterization of the adjoints of residuated ∧-homomorphisms (which need not preserve the top elements).

Theorem 4.4. A map f : B −→ A between implicative semilattices is →-semi-
linear, or equivalently, adjoint to a ∧-homomorphism, iff each basic closed subset of A has a basic closed preimage whose complement in T B is a subset of (but not necessarily equal to) the preimage of the complement in T A.
Proof. If f : B −→ A is adjoint to a ∧-homomorphism h : A −→ B then preimages of basic closed sets ca are basic closed: f ← ca = cha. By Proposition 3.8, aa is the complement ¬ ca of ca in T A, and by Proposition 4.1, we have That proper inclusion may occur is witnessed by Example 1.2.
Conversely, assume that f ← ca is basic closed and ¬ f ← ca ⊆ f ← ¬ ca for all a ∈ A. The first condition just expresses that f is adjoint to a map h : A −→ B with f ← ca = cha. From the second condition, it follows as above that Thus, by Proposition 4.1, f is → -semilinear, or equivalently, adjoint to a ∧homomorphism. A map f between topped posets is called codense if fb = implies b = . If f is adjoint to h then preservation of top elements by h is equivalent to codensity of f : If for an l-morphism f : A −→ B and some C ∈ SlA there is a greatest D ∈ SlB contained in the preimage f ← C then this D is called the localic preimage of C and denoted by f ← C. For the complete case, one finds the following result in [37] For categorically versed readers: the l-inclusion map of the localic preimage under a localic map f is the pullback of the l-inclusion map along f [38], and one defines localic preimages of extremal l-monomorphisms, regarded as sublocales, by taking pullbacks [21,22,39]. Non-complete situations are less comfortable, as is straightforward, using the fact that a B is injective. Vol. 83 (2022) Adjoint maps between implicative semilattices Note that a map f : B −→ A is an l-morphism iff it is codense and there exists any map h : A −→ B satisfying the Frobenius identity for → , because that entails To make the condition (e) in Theorem 4.6 more "symmetric", one may add that localic preimages of basic closed sets are basic closed. An obvious question is whether condition (d) is tantamount to the weaker condition It is true that any such map h has to be isotone on account of the implications and that h has to commute with all existing complements: However, condition (d') does not imply that f is isotone, not even if h is the identity map on an eight-element boolean algebra B: The sketched map f is extensive and idempotent but not isotone, and satisfies This example also shows that in condition (c) of Theorem 4.6 it does not suffice to postulate localic preimages of basic closed sets to be basic closed. But isotone maps may also be characterized by a continuity condition, namely with respect to the topologies formed by all unions of basic closed sets.
Let us summarize the main conclusions for the case of locales (where the l-domains are the sublocales) and stress the analogy but also the differences to the classical case of topological spaces. Applying Theorem 4.6 to the complete case shows the localic maps in a very pleasing light, namely as a natural analogue of the topologically continuous functions. In accordance with [37, Ch. III-4] we have for any localic map that

Biadjoint morphisms
By a biadjoint map between posets we mean one that is both adjoint and coadjoint. A biadjoint map preserves not only all existing joins, but also all existing meets. Hence, a biadjoint map between semilattices is certainly an r-morphism; and a map between complete lattices is biadjoint iff it preserves arbitrary joins and meets, in other words, it is a complete homomorphism.
Let us consider some further Frobenius identities: Proof. The claim is immediate from the following chains of equivalences: Thus, if h has a coadjoint g then the equivalence gb ≤ a ⇔ b ≤ ha yields f ab ⊆ aa ⇔ agb ⊆ aa, whence agb is the least basic open set containing f ab. Conversely, if such an agb exists for each b ∈ B, then we obtain i.e., g is coadjoint to h.
We are ready for a generalized version of the Joyal-Tierney Theorem [26] about open localic maps between frames/locales (cf. [37] for the sublocale version): Conversely, assuming that h is coadjoint to f , adjoint to g, and preserves →, we use (a) ⇒ (c) in Proposition 5.1 twice to prove f ab = agb, which will show that f is basic open. For d = b → d ∈ ab and a = fd we get ha ≤ d and Thus, f ab ⊆ agb. And each gb → c ∈ agb is equal to f (b → hc) ∈ f ab, whence agb ⊆ f ab.

Closure in Heyting lattices and interior in locales
The formation of closure and interior in topological spaces has strict analogues for frames/locales (but not for implicative semilattices, as certain completeness properties are required in order to guarantee the existence of the embedding c of A in NA and of localic preimages; see Propositions 3.8 and 4.5). Some of the results below are folklore in pointfree topology; the formulation via concrete sublocale sets (cf. [35]) makes the involved concepts more handy.
Let Note that for the l-domain D in Example 4.3, neither uD nor auD exists. Lemma 6.1. If A is a locale and C ∈ SlA has a complement ¬ C in SlA then With respect to closure and interior, localic maps between locales behave quite similar to but not completely like continuous maps between spaces. Indeed, from the equivalence (a) ⇔ (d) ⇔ (e) in Theorem 4.6 one easily derives a further characterization of localic maps in terms of continuity: An isotone map f : B −→ A between locales is localic iff the localic preimages f ← C of all sublocales C ∈ SlA exist and satisfy Conversely, if (f ← C) • ⊆ f ← C • for all C ∈ SlA then, since for each open sublocale ab of B the image f ab is a sublocale of A, we get In other words, f ab is open.
The following characterization of boolean l-ideals is given in [16] (for the complete case see [37]): Using this fact, we prove: Proof. By Theorem 5.3, h is adjoint to a map g. For all b ∈ B and c ∈ C, For c ∈ C, we have hc ∈ f ← C, hence b ≤ hc and a = gb ≤ c, that is, c ∈ ca. Thus, C ⊆ ca, C ⊆ ca, and so Summarizing the previous results, we arrive at the following closuretheoretical characterization of open localic maps: Theorem 6.6. An isotone map f : B −→ A between locales is localic and open iff the localic preimages f ← C of all sublocales C ∈ SlA exist and satisfy In contrast to the situation with spaces, a localic map f satisfying for all sublocales C need not be open, as the following reasoning shows: where − refers to A. Indeed, any a → ⊥ ∈ C = ↑⊥C must already be in C, since a → ⊥ ≥ ⊥C implies a → ⊥ = a → (a ∧ ⊥C) = a → ⊥C ∈ C.
For a thorough investigation of localic maps satisfying the above closure equation and related conditions (referred to as hereditarily skeletal maps) in a more categorical environment see Johnstone [25]. Vol. 83 (2022) Adjoint maps between implicative semilattices Page 19 of 23 13

Basic zero-dimensional spaces
Our results suggest to consider so-called basic zero-dimensional (closure) spaces. These are triples S = (X, C, D) where D is a closure system on X that is distributive as a lattice, C is a subset of D, X ∈ C, and each C ∈ C has a complement ¬ C in D, B = {B ∨ ¬ C | B, C ∈ C} is a meet-base of D , which means that By distributivity, complements in D coincide with the pseudocomplements and are therefore unique. We call the members of C basic closed and their complements basic open; but notice that C need not be a closure system. Putting we observe that S c = (|S|, AS, DS) is a basic zero-dimensional space, too, the complementary space of S; indeed, S cc = S.
A basic continuous map between basic zero-dimensional spaces S and T is a map f : |S| −→ |T | such that the preimage of DT is DS, preimages of basic closed sets are basic closed, and their lattice complements in DS are contained in the preimages of the complements in DT : After having checked the composition law for basic continuous maps, one obtains a category B0ds of basic zero-dimensional spaces.
Here are a few prominent instances.
(1) Each T D -closure space (X, C) (in which {x} {x} is closed for all x ∈ X, see [11], and for the topological case [2,37]) may be regarded as a basic zero-dimensional space (X, C, PX); indeed, C is a meet-base for PX on account of the equation X {x} = B ∪ (X C), where B = {x} {x} and C = {x} are in C. Then one checks that the category of T D -closure spaces with the usual continuous maps is fully embedded in B0ds. (2) More generally, consider any closure space (X, C) together with the topological closure system D consisting of all closed sets with respect to the topology generated by the differences C D with C, D ∈ C. The triple (X, C, D) is then a basic zero-dimensional space in which complements are formed set-theoretically. In the case of a topological closure system C, the topological space associated with (X, D) is known as the front space of (X, C) and its topology as the Skula topology ; see [37,44]. where D is the system of closed sets and C consist of all clopen sets, one obtains another category fully embedded in B0ds, namely that of zero-dimensional spaces and maps such that preimages of clopen sets are clopen-an important tool, e.g., in Stone duality. Recall that for boolean spaces (Stone spaces in [22]), that is, compact zero-dimensional Hausdorff spaces, the basic continuous maps are just the continuous ones. (4) If A is an implicative semilattice with underlying set X then, for the closure system MA generated by BA, the triple (X, cA, MA) is a basic zero-dimensional space. By Theorem 4.6, the l-morphisms are just the basic continuous maps between them. Thus, the category of implicative semilattices and l-morphisms is fully embedded in B0ds. Specifically, in boolean lattices, the fact that all l-domains are basic open and closed considerably simplifies the situation: here, MA is merely the MacNeille completion of A. On the other hand, in the case of frames, MA coincides with NA.
These examples may suffice for the moment to motivate future investigation of basic zero-dimensional spaces and suitable morphisms between them.
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