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The fact that equalizers in the context of strongly Hausdorff locales (similarly like those in classical spaces) are closed is a special case of a standard categorical fact connecting diagonals with general equalizers. In this paper we analyze this and related phenomena in the category of locales. Here the mechanism of pullbacks connecting equalizers is based on natural preimages that preserve a number of properties (closedness, openness, fittedness, complementedness, etc.). Also, we have a new simple and transparent formula for equalizers in this category providing very easy proofs for some facts (including the general behavior of diagonals). In particular we discuss some aspects of the closed case (strong Hausdorff property), and the open and clopen one.

A meet in a frame is exact if it join-distributes with every element, it is strongly exact if it is preserved by every frame homomorphism. Hence, finite meets are (strongly) exact which leads to the concept of an exact resp. strongly exact filter, a filter closed under exact resp. strongly exact meets. It is known that the exact filters constitute a frame $${\mathrm{Filt}}_{{\textsf {E}}}(L)$$ somewhat surprisingly isomorphic to the frame of joins of closed sublocales. In this paper we present a characteristic of the coframe of meets of open sublocales as the dual to the frame of strongly exact filters $${\mathrm{Filt}}_{{\textsf {sE}}}(L)$$ .

We prove, for a general frame, that the sublocales that can be represented as joins of closed ones are, somewhat surprisingly, in a natural one-to-one correspondence with the filters closed under exact meets, and explain some subfit facts from this perspective. Furthermore we discuss the filters associated in a similar vein with the fitted sublocales.

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.

It is shown that every d-frame admits a complete lattice of quotients. Quotienting may be triggered by a binary relation on one of the two constituent frames, or by changes to the consistency or totality structure, but as these are linked by the reasonableness conditions of d-frames, the result in general will be that both frames are factored and both consistency and totality are increased.

The property to be discussed in this chapter is very restrictive, so restrictive that one may hesitate to think about it as of an honest separation axiom (similarly like one does not usually think of discreteness as of a separation property: it says that distinct points can be separated by closed sets, but this is really somehow going too far).

We can only agree with Peter Johnstone who wrote in Johnstone (Bull Amer Math Soc (N.S.) 8:41–53, 1983) that

Here we start with two more variants of normality. There is the perfect normality, which turns out to be a conjunction of the classical perfectness (which is slightly different in the point-free context due to the different behaviour of sublocales and subspaces) and normality; in a way it can be viewed as a weaker form of metrizability. Next we deal with the technically important collectionwise normality. Then, in the penultimate section we prove and discuss the Katětov–Tong insertion theorem, using (to advantage) the techniques of the point-free real line. We finish with a certain duality between normality and extremal disconnectedness that allows to translate several results concerning normality to facts about extremal disconnected frames.

So far (with just a few exceptions) we have discussed the “low separation axioms”: those that can be used to replace the classical T1, and several conditions mimicking the Hausdorff property. Before turning to the stronger ones we will now summarize relations between them and add some facts that will help to understand the role they play and can play in further applications.

Point-free topology is the study of the category of locales and localic maps and its dual category of frames and frame homomorphisms. These notes cover the topics presented by the first author in his course on Frames and Locales at the Summer School in Algebra and Topology. We give an overview of the basic ideas and motivation for point-free topology, explaining the similarities and dissimilarities with the classical setting and stressing some of the new features.

In this final chapter, after briefly summarizing some of the already discussed facts concerning subfitness and fitness, we will tackle an aspect of these properties we have not examined yet, the role they play in the links of the phenomena of completeness, openness and the Heyting structure. Let us explain the main topic we will be interested in.

We will start by recalling some standard separation axioms in topological spaces X and discussing how to cope with them without points. In the point-free setting we think of classical spaces in terms of the complete lattices Ω(X) of their open sets, and of general spaces as of complete lattices of similar nature. Hence, the fact that the classical separation is expressed by statements in which individual points (and non-open subsets) play a prominent role seems to be a principal obstacle for extending them to the more general context. But the situation is much more favourable than what one may expect. First of all, some of the separation conditions can be replaced by obviously equivalent statements using the calculus of the lattice only: such is an absolutely straightforward reformulation of normality, very easy (and classically transparent) reformulation of regularity, and a translation of complete regularity which needs some more explanation (but this explanation concerns the role of real functions which calls for explanation in the classical situation as well). This will be presented already in this chapter. Then there are reformulations and replacements that are more involved (in particular the Hausdorff axiom that will come in variants with nontrivial relations). And then there are specific point-free separation conditions that are of particular interest: some of them akin to classical ones but not quite equivalent, some of them naturally arising from lattice theoretic requirements. They will be discussed and analysed in the individual chapters below (to be more exact: regularity, complete regularity and normality will also have their individual chapters; what we have in mind is that their reformulations can be presented right away while the others will be postponed).

Among the conditions of separation type, the axiom of regularity is very special. From the earliest stages of point-free topology there was natural interest in conditions that would capture classical separation phenomena as convincingly as possible in the new, more general, context. Classical separation axioms are typically formulated in the language of points and point-dependent notions; hence, one looked for equivalent formulations, or imitated the geometric intuition to obtain suitable replacements or at least analogies (see, e.g., Isbell (Math Scand 31:5–32, 1972), Dowker and Strauss. Separation axioms for frames. In: Topics in Topology, pp. 223–240. Proc. Colloq., Keszthely, 1972. Colloq. Math. Soc. Janos Bolyai, vol. 8, North-Holland, Amsterdam, 1974, Isbell (Math Scand 36:317–339, 1975), Simmons. A framework for topology. In: Logic Colloq. ’77, pp. 239–251. Stud. Logic Foundations Math., vol. 96. North-Holland, Amsterdam-New York, 1978, Johnstone. Stone Spaces. Cambridge Studies in Advanced Mathematics, vol. 3. Cambridge University Press, Cambridge 1982, Rosický and Smarda (Math. Proc. Cambridge Philos. Soc. 98:81–86, 1985)). In this company, regularity stands out. As we have already seen in Chap. I, it can be translated very easily, and the obtained formula has a natural appeal even in classical spaces (in fact, in an obviously equivalent form it is used classically anyway). There is no reasonable doubt that this formula makes a fully satisfactory point-free extension. It can be used without problems for proving useful facts parallel with the classical ones; moreover, it is algebraically versatile and easy to work with.

Exact meets in a distributive lattice are the meets ⋀ i a i such that for all b, ( ⋀ i a i ) ∨ b = ⋀ i ( a i ∨ b ) ; strongly exact meets in a frame are preserved by all frame homomorphisms. Finite meets are, trivially, (strongly) exact; this naturally leads to the concepts of exact resp. strongly exact filters closed under all exact resp. strongly exact meets. In [2] , [12] it was shown that the subsets of all exact resp. strongly exact filters are sublocales of the frame of up-sets on a frame, hence frames themselves, and, somewhat surprisingly, that they are isomorphic to the useful frame S c ( L ) of sublocales join-generated by closed sublocales resp. the dual of the coframe meet-generated by open sublocales. In this paper we show that these are special instances of much more general facts. The latter concerns the free extension of join-semilattices to coframes; each coframe homomorphism lifting a general join-homomorphism φ : S → C (where S is a join-semilattice and C a coframe) and the associated (adjoint) colocalic maps create a frame of generalized strongly exact filters (φ-precise filters) and a closure operator on C (and – a minor point – any closure operator on C is thus obtained). The former case is slightly more involved: here we have an extension of the concept of exactness (ψ-exactness) connected with the lifts of ψ : S → C with complemented values in more general distributive complete lattices C creating, again, frames of ψ-exact filters; as an application we learn that if such a C is join-generated (resp. meet-generated) by its complemented elements then it is a frame (resp. coframe) explaining, e.g., the coframe character of the lattice of sublocales, and the (seemingly paradoxical) embedding of the frame S c ( L ) into it.

We study uniformities and quasi-uniformities (uniformities without the symmetry axiom) in the common language of entourages. The techniques developed allow for a general theory in which uniformities are the symmetric part. In particular, we have a natural notion of Cauchy map independent of symmetry and a very simple general completion procedure (perhaps more transparent and simpler than the usual symmetric one).

Tightness is a notion that arose in an attempt to understand the reverse reflection problem: given a monoreflection of a category onto a subcategory, determine which subobjects of an object in the subcategory reflect to it — those which do are termed tight. Thus tightness can be seen as a strong density property. We present an analysis of λ-tightness, tightness with respect to the localic Lindel¨of reflection. Leading to this analysis, we prove that the normal, or Dedekind-MacNeille, completion of a regular σ-frame A is a frame. Moreover, the embedding of A in its normal completion is the Bruns-Lakser injective hull of A in the category of meet semilattices and semilattice homomorphisms.Since every regular σ-frame is the cozero part of a regular Lindel¨of frame, this result points towards λ-tightness. For any regular Lindel¨of frame L, the normal completion of Coz L embeds in L as the sublocale generated by Coz L. Although this completion is clearly contained in every sublocale having the same coz...

We present a new proof of Banaschewski's theorem stating that the completion lift of a uniform surjection is a surjection. The new procedure allows to extend the fact (and, similarly, the related theorem on closed uniform sublocales of complete uniform frames) to quasi-uniformities ("not necessarily symmetric uniformities"). Further, we show how a (regular) Cauchy point on a closed uniform sublocale can be extended to a (regular) Cauchy point on the larger (quasi-)uniform frame.

The frame Sc(L) generated by closed sublocales of a locale L is known to be a natural Boolean ("discrete") extension of a subfit L; also it is known to be its maximal essential extension. In this paper we first show that it is an essential extension of any L and that the maximal essential extensions of L and Sc(L) are isomorphic. The construction Sc is not functorial; this leads to the question of individual liftings of homomorphisms L → M to homomorphisms Sc(L) → Sc(M). This is trivial for Boolean L and easy for a wide class of spatial L, M. Then, we show that one can lift all h : L → 2 for weakly Hausdorff L (and hence the spectra of L and Sc(L) are naturally isomorphic), and finally present liftings of h : L → M for regular L and arbitrary Boolean M.Mathematics Subject Classification (2010): 06D22, 54D10, 54D35.Keywords: Frame, locale, sublocale, sublocale lattice, essential extension, subfit, Booleanization

The cuts of the classical Dedekind-MacNeille completion DM(S) of a meet semilattice S give rise to a natural cut coverage in the down-set frame \({\mathcal{D}S}\): down-set D covers element s if s lies below all upper bounds of D. This, in turn, leads to what we call the Dedekind-MacNeille frame extension DMF(S). The meet semilattices S for which DM(S) = DMF(S), which we refer to as proHeyting semilattices, can be specified by a simple formula, and we provide a number of equivalent characterizations. A sample result is that DM(S) = DMF(S) iff DM(S) is a Heyting algebra iff DM(S) coincides with the Bruns-Lakser injective envelope.

This paper contains some new facts about subfitness and weak subfitness. In the case of spaces, subfitness is compared with the axiom of symmetry, and certain seeming discrepancies are explained. Further, Isbell’s spatiality theorem in fact concerns a stronger form of spatiality (T 1-spatiality) which is compared with the T D -spatiality. Then, a frame is shown to be subfit iff it contains no non-trivial replete sublocale, and the relation of repleteness and subfitness is also discussed in spaces. Another necessary and sufficient condition for subfitness presented is the validity of the meet formula for the Heyting operation, which was so far known only under much stronger conditions.

It is well known that a locale is subfit iff each of its open sublocales is a join of closed ones, and fit iff each of its closed sublocales is a meet of open ones. This formulation, however, exaggerates the parallelism between the behavior of fitness and subfitness. For it can be shown that a locale is fit iff each of its sublocales is a meet of closed ones, but it is not the case that a locale is subfit iff each of its sublocales is a join of closed ones. Thus we are led to take up the very natural question of which locales have the feature that every sublocale is a join of closed sublocales. In this note we show that these are precisely the subfit locales which are scattered in the pointfree sense of (13), and we add a variation for spatial frames.

We address what might be termed the reverse reflection problem: given a monoreflection from a category A onto a subcategory B , when is a given object B ∈ B the reflection of a proper subobject? We start with a well known specific instance of this problem, namely the fact that a compact metric space is never the Cech-Stone compactification of a proper subspace. We show that this holds also in the pointfree setting, i.e., that a compact metrizable locale is never the Cech-Stone compactification of a proper sublocale. This is a stronger result than the classical one, but not because of an increase in scope; after all, assuming weak choice principles, every compact regular locale is the topology of a compact Hausdorff space. The increased strength derives from the conclusion, for in general a space has many more sublocales than subspaces. We then extend the analysis from metric locales to the broader class of perfectly normal locales, i.e., those whose frame of open sets consists entirely of cozero elements. We include a second proof of these results which is purely algebraic in character. At the opposite extreme from these results, we show that an extremally disconnected locale is a compactification of each of its dense sublocales. Finally, we analyze the same phenomena, also in the pointfree setting, for the 0-dimensional compact reflection and for the Lindel o f reflection. Keywords: Cech-Stone compactification, regular locale, cozero element

We show that every frame can be essentially embedded in a Boolean frame, and that this embedding is the maximal essential extension of the frame in the sense that it factors uniquely through any other essential extension. This extension can be realized as the embedding $$L \rightarrow \mathcal {N}(L) \rightarrow \mathcal {B}\mathcal {N}(L)$$ , where $$L \rightarrow \mathcal {N}(L)$$ is the familiar embedding of L into its congruence frame $$\mathcal {N}(L)$$ , and $$\mathcal {N}(L) \rightarrow \mathcal {B}\mathcal {N}(L)$$ is the Booleanization of $$\mathcal {N}(L)$$ . Finally, we show that for subfit frames the extension can also be realized as the embedding $$L \rightarrow {{\mathrm{S}}}_\mathfrak {c}(L)$$ of L into its complete Boolean algebra $${{\mathrm{S}}}_\mathfrak {c}(L)$$ of sublocales which are joins of closed sublocales.

Sublocales of a locale (frame, generalized space) can be equivalently represented by frame congruences. In this paper we discuss, a.o., the sublocales corresponding to complete congruences, that is, to frame congruences which are closed under arbitrary meets, and present a “geometric” condition for a sublocale to be complete. To this end we make use of a certain closure operator on the coframe of sublocales that allows not only to formulate the condition but also to analyze certain weak separation properties akin to subfitness or $$T_1$$ . Trivially, every open sublocale is complete. We specify a very wide class of frames, containing all the subfit ones, where there are no others. In consequence, e.g., in this class of frames, complete homomorphisms are automatically Heyting.

Products of locales (generalized spaces) are coproducts of frames. Because of the algebraic nature of the latter they are often viewed as algebraic objects without much topological connotation. In this paper we first analyze the frame construction emphasizing its tensor product carrier. Then we show how it can be viewed topologically, that is, in the sum-of-the-open-rectangles perspective. The main aim is to present the product from different points of view, as an algebraic and a geometric object, and persuade the reader that both of them are fairly transparent.

Generalizing the obvious representation of a subspace $${Y \subseteq X}$$ as a sublocale in Ω(X) by the congruence $${\{(U, V ) | U\cap Y = V \cap Y\}}$$ , one obtains the congruence $${\{(a, b) |\mathfrak{o}(a) \cap S = \mathfrak{o}(b) \cap S\}}$$ , first with sublocales S of a frame L, which (as it is well known) produces back the sublocale S itself, and then with general subsets $${S\subseteq L}$$ . The relation of such S with the sublocale produced is studied (the result is not always the sublocale generated by S). Further, we discuss in general the associated adjunctions, in particular that between relations on L and subsets of L and view the aforementioned phenomena in this perspective.

Minimal obstructions to full homomorphisms to a graph B have been proved to be of size at most |B|+1. This turns out to require that disconnected obstructions be allowed. In this paper we prove that the size of minimal connected obstructions is at most |B|+2. We also prove that achieving |B|+2 is rare and present a complete list of the exceptional cases. Finally, we compute the dualities associated with these exceptions.

Each meet semilattice S is well known to be freely extended to a frame by its down-sets DS. In this article we present, first, the complete range of frame extensions generated by S; it turns out to be a sub-coframe of the coframe C of sublocales of DS, indeed, an interval in C, with DS as the top and the extension of S respecting all the exact joins in S as the bottom. Then, the Heyting and Boolean case is discussed; there, the bottom extension is shown to coincide with the Dedekind-MacNeille completion. The technique used is a technique of sites, generalizing that used in [JOHNSTONE, P. T.: Stone Spaces. Cambridge Stud. Adv. Math. 3, Cambridge University Press, Cambridge, 1982].

The concepts of fitness and subfitness (as defined in Isbell, Trans. Amer. Math. Soc. 327, 353–371, 1991) are useful separation properties in point-free topology. The categorical behaviour of subfitness is bad and fitness is the closest modification that behaves well. The separation power of the two, however, differs very substantially and subfitness is transparent and turns out to be useful in its own right. Sort of supplementing the article (Simmons, Appl. Categ. Struct. 14, 1–34, 2006) we present several facts on these concepts and their relation. First the “supportive” role subfitness plays when added to other properties is emphasized. In particular we prove that the numerous Dowker-Strauss type Hausdorff axioms become one for subfit frames. The aspects of fitness as a hereditary subfitness are analyzed, and a simple proof of coreflectivity of fitness is presented. Further, another property, prefitness, is shown to also produce fitness by heredity, in this case in a way usable for classical spaces, which results in a transparent characteristics of fit spaces. Finally, the properties are proved to be independent.

In the original publication of the article, the formulation of the c-subfitness condition (c-sfit) in Subsection 5.2 is inaccurate, with effect in Theorem 5.3.

An exact meet in a lattice is a special type of infimum characterized by, inter alia, distributing over finite joins. In frames, the requirement that a meet is preserved by all frame homomorphisms makes for a slightly stronger property. In this paper these concepts are studied systematically, starting with general lattices and proceeding through general frames to spatial ones, and finally to an important phenomenon in Scott topologies.

Let κ Qnt be the category of κ-quantales, quantales closed under κ-joins in which the monoid identity is the largest element. (κ is an infinite regular cardinal.) Although the lack of lattice completeness in this setting would seem to mitigate against the techniques which lend themselves so readily to the calculation of frame quotients, we show how to easily compute κ Qnt quotients by applying generalizations of the frame techniques to suitable extensions of this category. The second major tool in the analysis is the free κ-quantale over a λ-quantale, κ ⩾ λ . Surprisingly, these can be characterized intrinsically, and the generating sub-κ-quantale can even be identified. The result that the λ-free κ-quantales coincide with the λ-coherent κ-quantales directly generalizes Maddenʼs corresponding result for κ-frames. These tools permit a direct and intuitive construction of κ Qnt colimits. We provide two applications: an intrinsic characterization of κ Qnt colimits, and of free (over sets) κ-quantales. The latter is a direct generalization of Whitmanʼs condition for distributive lattices.

Due to the nature of product in the category of locales, the entourage uniformities in the point-free context only mimic the classical Weil approach while the cover (Tukey type) ones can be viewed as an immediate extension. Nevertheless the resulting categories are concretely isomorphic. We present a transparent construction of this isomorphism, and apply it to the natural uniformities of localic groups. In particular we show that localic group homomorphisms are uniform, thus providing natural forgetful functors from the category of localic groups into any of the two categories of uniform locales.

Let Y and P be posets, let P be finite and connected, and let f : Y → P be a surjective monotone map. The map f can be naturally extended to a Priestley surjection \({\widehat{f} : \widehat{Y} \to P}\) which can turn out to be a retraction even if f is not. We characterize those maps f whose Priestley extensions \({\widehat{f}}\) are retractions.

We investigate Heyting varieties determined by prohibition of systems of configurations in Priestley duals; we characterize the configuration systems yielding such varieties. On the other hand, the question whether a given finitely generated Heyting variety is obtainable by such means is solved for the special case of systems of trees.

Click on the link to view the abstract. Keywords: epimorphisms; epicomplete; episurjective; metric frames; contractions Quaestiones Mathematicae 31(2008), 241–253

A colored graph is a complete graph in which a color has been assigned to each edge, and a colorful cycle is a cycle in which each edge has a different color. We first show that a colored graph lacks colorful cycles iff it is Gallai, i.e., lacks colorful triangles. We then show that, under the operation $m\circ n\equiv m+n-2$, the omitted lengths of colorful cycles in a colored graph form a monoid isomorphic to a submonoid of the natural numbers which contains all integers past some point. We prove that several but not all such monoids are realized. We then characterize exact Gallai graphs, i.e., graphs in which every triangle has edges of exactly two colors. We show that these are precisely the graphs which can be iteratively built up from three simple colored graphs, having $2$, $4$, and $5$ vertices, respectively. We then characterize in two different ways the monochromes, i.e., the connected components of maximal monochromatic subgraphs, of exact Gallai graphs. The first characterization is in terms of their reduced form, a notion which hinges on the important idea of a full homomorphism. The second characterization is by means of a homomorphism duality.

Click on the link to view the abstract. Keywords: Prime and covered prime elements in frames, complete frame homomorphisms, well-ordered frame Quaestiones Mathematicae 37(2014), 451-454

Prohibiting configurations (≡ induced finite connected posets) in Priestley spaces and properties of the associated classes of distributive lattices, and the related problem of configurations in coproducts of Priestley spaces, have been brought to satisfactory conclusions in case of configurations with a unique maximal element. The general case is, however, far from settled. After a short survey of known results we present the desired answers for a large (although still not complete) class of configurations without top.

A natural structure modelling approximation is presented and the resulting category is shown to be equivalent with the category of complete nearness frames.