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AbstractWe study a variant of extremal disconnectedness in frames, defined by requiring that any two disjoint open sublocales should be separable by disjoint open sublocales each of which is the interior of some zero-sublocale. In spaces, this notion was recently introduced in [1] in order to study some properties of frames of z-ideals of rings of continuous functions. The point-free approach adopted here brings to the fore some topological results not considered in [1], such as that a dense z-embedded subspace of a Tychonoff space has this property if and only if the space containing it has the property. In [1], this was proved only for the embedding X ⊆ βX. We give a ring-theoretic characterization by defining a property of f-rings in a transparent manner which shows why the property should be defined the way we do. The property is that if two annihilator ideals meet trivially, then the annihilator of each of the annihilator ideals contains a positive element such that the sum of the two elements is a non-zerodivisor. [ABSTRACT FROM AUTHOR]

The dissolution (introduced by Isbell in [3], discussed by Johnstone in [5] and later exploited by Plewe in [12, 13]) is here viewed as the relation of the geometry of L with that of the more dispersed T(L) = S(L)op mediated by the natural embedding and its adjoint localic map γL: T(L) → L. The associated image-preimage adjunction (γL)−1 [−] ⊣ γL [−] between the frames T(L) and TT(L) is shown to coincide with the adjunction cT(L) ⊣ γT(L) of the second step of the assembly (tower) of L. This helps to explain the role of T(L) = S(L)op as an "almost discrete lift" (sometimes used as a sort of model of the classical discrete lift DL → L) as a dispersion going halfway to Booleanness. Consequent use of the concrete sublocales technique simplifies the reasoning. We illustrate it on the celebrated Plewe's theorem on ultranormality (and ultraparacompactness) of S(L) which becomes (we hope) substantially more transparent. [ABSTRACT FROM AUTHOR]

We offer here a particularly simple and direct construction of the congruence frame of a given frame, aka its assembly, from the bounded meet semilattice of differences of frame elements. The construction enables economical proofs of two of the assembly's known attributes, namely ultranormality and ultraparacompactness, as well as a proof that the assembly is an essential extension of the frame with the same essential closure as the frame. The major new result is that the assembly is free over its meet semilattice of differences. [ABSTRACT FROM AUTHOR]

The property of subfitness used in point-free topology (roughly speaking) to replace the slightly stronger T1-separation, appeared (as disjunctivity) already in the pioneering Wallman's [16], then practically disappeared to reappear again (conjunctivity, subfitness), until it was in the recent decades recognized as an utmost important condition playing a very special role. Recently, it was also observed that this property (or its dual) appeared independently in general poset setting (e.g. as separativity in connection with forcing). In a recent paper [2], Delzell, Ighedo and Madden discussed it in the context of semilattices. In this article we discuss it on the background of the systems of meet-sets (subsets closed under existing infima) in posets of various generality (semilattices, lattices, distributive lattices, complete lattices) and present parallels of some localic (frame) facts, including a generalized variant of fitness. [ABSTRACT FROM AUTHOR]

This paper considers the compact hedgehog as a frame presented by generators and relations, based on the presentation of the frame of extended real numbers. The main focus will be on the point-free version of continuous and semi-continuous functions that arise from it, and their application in characterizations of variants of collectionwise normality. The variants to be considered are defined by selections of adequate families of sublocales and their characterizations depend on lattice-theoretic properties of the selected families. This way insertion and extension results for semicontinuous and continuous functions with values in the compact hedgehog frame are generalized and unified. [ABSTRACT FROM AUTHOR]

This paper concerns the congruence frame in the setting of partial frames, which, in contrast to full frames, do not necessarily have all joins. Examples of these include bounded distributive lattices, σ- and κ-frames. A new class of congruences, called Heyting congruences, helps to illuminate the structure of these congruence frames. Defining these involves the very useful free frame over a partial frame. We investigate the relationship between Heyting congruences and co-atoms of the congruence frame, showing how different the cases of full and partial frames can be. The Madden congruence, which yields the least dense quotient, is recognized as a particular Heyting congruence. While the collection of all Heyting congruences on a partial frame is in general not even a meet-semilattice, joining each of them with the Madden congruence results in a collection with considerably more interesting structure. [ABSTRACT FROM AUTHOR]

Let P be a prime ideal of the ring of continuous real-valued functions on a completely regular frame L, i.e., L. We study many new results about the residue class domains L/P with an emphasis on determining when the ordered L/P is a valuation domain (i.e., when given any two non-zero elements of L/P , one divides the other). A prime ideal P of L is called a valuation prime ideal if L/P is a valuation domain. A frame L is called an SV-frame if every prime ideal of L is a valuation prime ideal. We introduce and study two new generalizations of the SV-frames. The first is that of a quasi SV-frame in which every real maximal ideal of L that is not a minimal prime ideal contains a non-maximal prime ideal P such that L/P is a valuation domain. In the second, we define a frame L to be an almost SV-frame if every maximal ideal of L contains a minimal valuation prime ideal. A point I ∈ Pt(βL) is called a special βF -point if OI = {δ ∈ L : coz δ ∈ I} is a valuation prime ideal of L. It is shown that I is a special βF -point if and only if the pseudo-prime ideals of L containing OI that are not primary form a chain under set inclusion. [ABSTRACT FROM AUTHOR]

Sublocales of frames, even those representing subspaces (induced sublo-cales), are typically not complemented in the lattice of all sublocales. We present a necessary and sufficient condition for an induced sublocale to be so, and prove that all induced sublocales are complemented iff the space in question is hereditarily irresolvable, a property slightly weaker than – and in a broad class of spaces equivalent with – scatteredness (under which condition, by Simmons' result all sublocales are complemented). [ABSTRACT FROM AUTHOR]

Let C(X) denote the ring of all real-valued continuous functions on a topological space X; and C∞(X) be the subring of all functions C(X) which vanish at infinity. In [2], the paper "Rings of continuous functions vanishing at infinity," (Comment. Math. Univ. Carolin.45(3) (2004), 519–533), by A.R. Aliabad, F. Azarpanah, and M. Namdari, it is shown that for every completely regular Hausdorff space X, whenever C∞(X) ≠ (0), then there exists a locally compact space Y such that C∞(X) ≅ C∞(Y). In fact, the space Y may be considered as a nonempty open locally compact subspace of X. In the present paper, analogous results are derived in a pointfree context in which topological spaces are replaced by frames. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The notion of σ∗-properness of a subset of a frame is introduced. Using this notion, we give necessary and sufficient conditions for a frame to be weakly Lindelöf. We show that a frame is weakly Lindelöf if and only if its semiregularization is weakly Lindelöf. For a completely regular frame L, we introduce a condition equivalent to weak realcompactness based on maximal ideals of the cozero part of L. This enables us to show that every weakly realcompact almost P -frame is realcompact. A new characterization of weakly Lindelöf frames in terms of neighbourhood strongly divisible ideals of ᎡᏞ is provided. The closed ideals of ᎡᏞ equipped with the uniform topology are applied to describe weakly Lindelöf frames. [ABSTRACT FROM AUTHOR]

In this paper we model discontinuous extended real functions in pointfree topology following a lattice-theoretic approach, in such a way that, if L is a subfit frame, arbitrary extended real functions on L are the elements of the Dedekind-MacNeille completion of the poset of all extended semicontinuous functions on L. This approach mimicks the situation one has with a T1-space X, where the lattice F̄(X) of arbitrary extended real functions on X is the smallest complete lattice containing both extended upper and lower semicontinuous functions on X. Then, we identify real-valued functions by lattice-theoretic means. By construction, we obtain definitions of discontinuous functions that are conservative for T1-spaces. We also analyze semicontinuity and introduce definitions which are conservative for T0-spaces. [ABSTRACT FROM AUTHOR]

Point-free modeling of mappings that are not necessarily continuous has been so far based on the extension of a frame to its frame of sublocales, mimicking the replacement of a topological space by its discretization. This otherwise successful procedure has, however, certain disadvantages making it not quite parallel with the classical theory (see Introduction). We mend it in this paper using a certain extension Sc(L) of a frameL, which is, a.o., Boolean and idempotent. Doing this we do not lose the merits of the previous approach. In particular we show that it yields the desired results in the treatment of semicontinuity. Also, there is no obstacle to using it as a basis of a point-free theory of rings of real functions; the “ring of all real functions” F(L) = C(Sc(L)) is now order complete. [ABSTRACT FROM PUBLISHER]

We provide the appropriate unifying framework for the various descriptions of the Dedekind completion of the ring C(L) of continuous real functions on a frameL. It is based on suitable Galois connections and a general result about Galois connections, showing once more the ubiquity of (Galois) adjunctions between partially ordered sets and their conceptual simplicity and effectiveness. [ABSTRACT FROM PUBLISHER]

A partial frame is a meet-semilattice in which certain designated subsets are required to have joins, and finite meets distribute over these. The designated subsets are specified by means of a so-called selection function, denoted byS; these partial frames are calledS-frames. We construct free frames over S-frames using appropriate ideals, calledS-ideals. TakingS-ideals gives a functor fromS-frames to frames. Coupled with the functor from frames toS-frames that takesS-Lindelöf elements, it provides a category equivalence betweenS-frames and a non-full subcategory of frames. In the setting of complete regularity, we provide the functor takingS-cozero elements which is right adjoint to the functor takingS-ideals. This adjunction restricts to an equivalence of the category of completely regularS-frames and a full subcategory of completely regular frames. As an application of the latter equivalence, we construct the Stone-Č ech compactification of a completely regularS-frame, that is, its compact coreflection in the category of completely regularS-frames. A distinguishing feature of the study of partial frames is that a small collection of axioms of an elementary nature allows one to do much that is traditional at the level of frames or locales and of uniform or nearness frames. The axioms are sufficiently general to include as examples of partial frames bounded distributive lattices, σ-frames, κ-frames and frames. [ABSTRACT FROM PUBLISHER]

LetRLdenote the ring of continuous real-valued functions on a com- pletely regular frameL. The support of anα∈RLis the closed quotient ↑(cozα)∗. We show that if supports are coz-quotients inL, then the set of functions with realcompact support is an ideal. IfLsatisfies the stronger condition that supports areC-quotients, then this ideal is the intersection of pure parts of the free maximal ideals ofRL. The set of functions whose cozeroes are realcompact is always an ideal, which is free if and only ifLis locally realcompact if and only ifLis (isomorphic to) an open quotient ofυL. Further, this ideal is prime if and only if it is a free real maximal ideal if and only ifυL→Lis a one-point extension ofL. [ABSTRACT FROM AUTHOR]

We define Čech-complete frames by means of a filter condition which does not require that such frames be completely regular. We then, among regular frames, give a characterization in terms of ideals from which one sees more easily that a Tychonoff space is Čech-complete iff its frame of open sets is Čech-complete. Extending this notion of Čech-completeness to nearness frames in a natural way (meaning that covers are replaced with uniform covers, and convergent filters – the ones that meet every cover – are replaced with Cauchy filters – the ones that meet every uniform cover), we define controlled nearness frames along the lines that Bentley and Hunsaker define controlled nearness spaces. We show that the subcategory they form is closed under countable coproducts. [ABSTRACT FROM AUTHOR]

Completeness for metric spaces is traditionally presented in terms of convergence of Cauchy sequences, and for uniform spaces in terms of Cauchy filters. Somewhat more abstractly, a uniform space is complete if and only if it is closed in every uniform space in which it is embedded, and so isomorphic to any space in which it is densely embedded. This is the approach to completeness used in the point-free setting, that is, for uniform and nearness frames: a nearness frame is said to be complete if every strict surjection onto it is an isomorphism. Quasi-uniformities and quasi-nearnesses on biframes provide appropriate structures with which to investigate uniform and nearness ideas in the asymmetric context. In [9] a notion of completeness (called “quasi-completeness”) was presented for quasi-nearness biframes in terms of suitable strict surjections being isomorphisms, and a quasi-completion was constructed for any quasi-nearness biframe. In this paper we show that quasi-completeness can indeed be viewed in terms of the convergence of certain filters, namely, the regular Cauchy bifilters. We use the notion of aT-valued bifilter, which generalizes the characteristic function of a filter. An important tool is an appropriate composition for such bifilters. We show that the right adjoint of the quasi-completion is the universal regular Cauchy bifilter and use it to prove this characterization of quasi-completeness. We also construct the so-called Cauchy filter quotient for a biframe using a quotient of the downset biframe that involves only the Cauchy, and not the regularity, condition. Like the quasi-completion, this provides a universal Cauchy bifilter; unlike the quasi-completion, this construction is functorial. [ABSTRACT FROM AUTHOR]

Quasi-completeness was considered in [16], where a quasi-completion was constructed for any quasi-nearness biframe. In this paper, we compare the familiar notions of compactness and total boundedness with quasi-completeness. We investigate the uniqueness of quasi-completions, using the full and balanced biframes of [17]. In the case of quasi-uniform biframes it makes sense to compare this quasi-completion with the completion of [13] and [14] – surprisingly, they need not coincide. [ABSTRACT FROM AUTHOR]

By extending the works of Arias et al. and Balazs we present a natural sufficient condition via frames for the membership of the Schatten p-class operators for 1 ≤ p ≤ 2, and show that such a condition is not sufficient to guarantee the membership of the Schatten-class operators for p > 2. [ABSTRACT FROM AUTHOR]

In this note we present a new treatment of the pointfree version of the semicontinuous quasi-uniformity based on the new tool of the ring of arbitrary (not necessarily continuous) real-valued functions made available recently by J. Gutierrez Garcia, T. Kubiak and J. Picado [13]. Our purpose is to show how the basic facts about the semicontinuous quasi-uniformity can be nicely presented and proved with that tool at hand. [ABSTRACT FROM AUTHOR]

The paper provides another approach to the notion of variable-basis topological system generalizing the fixed-basis concept of S. Vickers, considers functorial relationships between the categories of modified variable-basis topological systems and variable-basis fuzzy topological spaces in the sense of S.E. Rodabaugh and shows that the procedure of localification is possible in the new setting. [ABSTRACT FROM AUTHOR]

We introduce quasi-nearness biframes. They provide a generalization of both nearness frames [5] and quasi-uniform biframes [12]. Quasi-nearness biframes are regular; they are normal if and only if the standard binary quasi-nearness is in fact a quasi-uniformity. Further, we introduce a notion of completeness, and provide a construction of a completion in which we make use of the new notion of a part-preserving right adjoint of a biframe map. [ABSTRACT FROM AUTHOR]

Considering L a frame, we introduce the notion of stratified L-neighbor- hood topological ring, produce some characterization theorems including its L- uniformizability. With the help of the notions of stratified convergence structures attributed to Gunther Jäger [10], we introduce and study various subcategories of stratified L-convergence rings. In doing so, we bring into light, among others, the notions of stratified L-uniform group and stratified L-uniform convergence group in an attempt to show that every stratified L-convergence ring carries in a natural way the stratified L-uniform convergence structure of Jäger and Burton [15], and the category of stratified L-uniform groups and uniformly continuous group homomorphisms, SL- UnifGrp is isomorphic to the category of principal stratified L-uniform convergence groups and uniformly continuous group homomorphisms, SL-PUConvGrp. Intro- ducing the notion of stratified L-Cauchy ring, we show that the category of stratified L-Cauchy rings and Cauchy-continuous ring homomorphisms, SL-ChyRng is topological over the category of rings, Rng with respect to the forgetful functor, and that every stratified L-Cauchy ring is a stratified L-convergence ring. We observe that if L is a Boolean algebra, then each stratified L-uniform convergence ring serves as a natural example of a stratified L-Cauchy ring. We give necessary and sufficient conditions for a stratified L-convergence structure and a ring structure to be a stratified L-convergence ring. [ABSTRACT FROM AUTHOR]

We give several characterizations of quasi F-frames. To name a few, let RL denote, as usual, the ring of real-valued continuous functions on a completely regular frame L. Then, L is quasi-F iff Ann²(α) + Ann²(β) = RL whenever α + β is not a zero-divisor. A commutative ring is said to be quasi-Bézout if every finitely generated ideal that contains a non-zero-divisor is principal. We show that L is quasi-F iff RL is quasi-Bézout. Further, L is quasi-F iff βL (its Stone-Čech compactification) is quasi-F, iff λL (its universal Lindelöfication) is quasi-F, iff υL (its Hewitt realcompactification) is quasi-F. Element-wise characterizations include one stating that a completely regular frame is quasi-F iff whenever the join of two cozero elements is dense, then their pseudocomplements are completely separated. [ABSTRACT FROM AUTHOR]

Oz frames are the natural pointfree counterpart of Oz spaces, that is, those topological spaces in which every open set is z-embedded. We give here the point-free analogues of known characterisations of Oz spaces and show that the Lindelöf coreflection of an Oz frame is Oz. The latter result has no spatial version, but has implications for a number of well-known frame and spatial extensions, characterizations and properties of these frames. Extremally disconnected frames are investigated in relation to Oz frames and weak Oz frames, the latter being a very natural generalisation of Oz. [ABSTRACT FROM AUTHOR]

Following the notion of stratified L-fuzzy convergence space of Gunther Jäger [Quaest. Math. 24 (2001), 501-517], we introduce the notion of a stratified L-generalized convergence group, and look into some other objects, namely, stratified L-Kent convergence groups, and stratified L-principal convergence groups. We show that the category of stratified L-generalized convergence groups, SL-GCGrp is topological over the category of groups, Grp with respect to the forgetful functor, and we prove that the category SL-NeighGrp, of stratified L-neighborhood groups is isomorphic to a subcategory of SL-GCGrp. We give necessary and sufficient conditions for a group structure and a stratified L-generalized convergence structure to be a stratified L-generalized convergence group. Finally, we observe that every stratified L-generalized convergence group possessing a stratified L-principal convergence structure gives rise to a stratified L-neighborhood topological group. [ABSTRACT FROM AUTHOR]