An asymmetric characterization of the congruence frame

The functor from regular biframes to frames, taking first parts, is shown to be faithful. This result is used to provide many examples of identical embeddings which are epimorphisms in the category of frames. Then the congruence frame, regarded as a biframe, is characterized as being the unique regular biframe extension. This provides a pointfree analogue to a result of Salbany (1970, 1974 [16]) that the forgetful functor from completely regular bitopological spaces to all topological spaces, taking first parts, has a unique section.     

An asymmetric approach to filters in strict extensions and quotients

Filters are a fundamental tool in the study of convergence and completeness in topology. On the other hand, downsets have been used extensively for this purpose in the setting of pointfree topology. This paper investigates links between these in the asymmetric context, that is, for biframes and bispaces. We present an appropriate kind of filter for the asymmetric setting, which we call a bifilter. These form a bispace, functorially so, which proves isomorphic to the spectrum of the downset biframe. As a corollary, downset biframes are seen to be isomorphic to the opens of the bifilter bispace. Both these correspondences are natural isomorphisms. The join map from a downset biframe to its underlying biframe appears here as a universal strict quotient. We use it to show that the embedding of any T ₀ bispace in its bifilter bispace is a universal strict extension. Banaschewski and Hong [10] have established the importance of general filters for convergence and completeness in the pointfree setting. We conclude this paper with a discussion of an appropriate concept of general bifilter, and show that the right adjoint of the join map mentioned above is a universal general bifilter.     

STRONG ZERO-DIMENSIONALITY OF BIFRAMES AND BISPACES

Abstract

A bispace is called strongly zero-dimensional if its bispace Stone—?ech compactification is zero—dimensional. To motivate the study of such bispaces we show that among those functorial quasi—uniformities which are admissible on all completely regular bispaces, some are and others are not transitive on the strongly zero-dimensional bispaces. This is in contrast with our result that every functorial admissible uniformity on the completely regular spaces is transitive precisely on the strongly zero-dimensional spaces.

We then extend the notion of strong zero-dimensionality to frames and biframes, and introduce a De Morgan property for biframes. The Stone—Cech compactification of a De Morgan biframe is again De Morgan. In consequence, the congruence biframe of any frame and the Skula biframe of any topological space are De Morgan and hence strongly zero-dimensional. Examples show that the latter two classes of biframes differ essentially.     

BIFRAMES AND BISPACES

Abstract

The concept of a biframe is introduced. Then the known dual adjunction between topological spaces and frames (i.e. local lattices) is extended to one between bispaces (i.e. bitopological spaces) and biframes. The largest duality contained in this dual adjunction defines the sober bispaces, which are also characterized in terms of the sober spaces. The topological and the frame-theoretic concepts of regularity, complete regularity and compactness are extended to bispaces and biframes respectively. For the bispaces these concepts are found to coincide with those introduced earlier by J.C. Kelly, E.P. Lane, S. Salbany and others. The Stone-?ech compactification (compact regular coreflection) of a biframe is constructed without the Axiom of Choice.     

Enough Regular Cauchy Filters for Asymmetric Uniform and Nearness Structures

Quasi-nearness biframes provide an asymmetric setting for the study of nearness; in Frith and Schauerte (Quaest Math 33:507–530, 2010) a completion (called a quasi-completion) was constructed for such structures and in Frith and Schauerte (Quaest Math, 2012) completeness was characterized in terms of the convergence of regular Cauchy bifilters. In this paper questions of functoriality for this quasi-completion are considered and one sees that having enough regular Cauchy bifilters plays an important rôle. The quasi-complete strong quasi-nearness biframes with enough regular Cauchy bifilters are seen to form a coreflective subcategory of the strong quasi-nearness biframes with enough regular Cauchy bifilters. Here a significant difference between the symmetric and asymmetric cases emerges: a strong (even quasi-uniform) quasi-nearness biframe need not have enough regular Cauchy bifilters. The Cauchy filter quotient leads to further characterizations of those quasi-nearness biframes having enough regular Cauchy bifilters. The fact that the Cauchy filter quotient of a totally bounded quasi-nearness biframe is compact shows that any totally bounded quasi-nearness biframe with enough regular Cauchy bifilters is in fact quasi-uniform. The paper concludes with various examples and counterexamples illustrating the similarities and differences between the symmetric and asymmetric cases.     

On Strong Inclusions and Asymmetric Proximities in Frames

The strong inclusion, a specific type of subrelation of the order of a lattice with pseudocomplements, has been used in the concrete case of the lattice of open sets in topology for an expedient definition of proximity, and allowed for a natural pointfree extension of this concept. A modification of a strong inclusion for biframes then provided a pointfree model also for the non-symmetric variant. In this paper we show that a strong inclusion can be non-symmetrically modified to work directly on frames, without prior assumption of a biframe structure. The category of quasi-proximal frames thus obtained is shown to be concretely isomorphic with the biframe based one, and shown to be related to that of quasi-uniform frames in a full analogy with the symmetric case.     

NIL PROBLEMS REVISITED FOR ALMOST NILPOTENT RINGS

Abstract

We begin with the notion of K-flat projectivity. For each biframe L we then introduce a binary relation ?L on it. The K-flat projective biframes are exactly such biframes with each element a of the total (first, second) part approximated by the elements x of the total (first, second) part, x?_L a and the relation ?_L being stable wrt. the meet operation on L. Further on, we introduce the notion of a K-comonad and characterize K-flat projective biframes as those biframes having a coalgebra structure for the K-comonad. The K-coherent biframes and K-flat projective biframes are coreflective in all biframes.     

Smooth spaces versus continuous spaces in models for synthetic differential geometry

In topos models for synthetic differential geometry we study connections between smooth spaces (which interpret synthetic calculus) and continuous spaces (which interpret intuitionistic analysis). Our main tools are adjoint retractions of toposes and the standard map from the smooth reals to the continuous reals.     

Straight nearness spaces

Straight spaces are spaces for which a continuous map defined on the space which is uniformly continuous on each set of a finite closed cover is then uniformly continuous on the whole space. Previously, straight spaces have been studied in the setting of metric spaces. In this paper, we present a study of straight spaces in the more general setting of nearness spaces. In a subcategory of nearness spaces somewhat more general than uniform spaces, we relate straightness to uniform local connectedness. We investigate category theoretic situations involving straight spaces. We prove that straightness is preserved by final sinks, in particular by sums and by quotients, and also by completions.     

The Samuel compactification for quasi-uniform biframes

The paircover approach is used to explore the links between quasi-uniform and proximal biframes. The Samuel compactification for quasi-uniform biframes is constructed and its universal property discussed.     

Cover quasi-uniformities in frames

Quasi-uniformities (not necessarily symmetric uniformities) are usually studied via entourages (special neighbourhoods of the diagonal in X×X) where one can simply forget about the symmetry requirement. This has been done successfully in the point-free context as well, but there is a demand for a covering approach, a.o. because the point-free representation of the square X×X is not without difficulties. Based on the (spatial) ideas from Gantner and Steinlage (1972) [9], a cover type quasi-uniformity was developed in Frith (1987) [6] and other papers using biframes, the point-free variant of bitopologies. In this paper we show that this can be avoided and present a cover type quasi-uniformity structure enriching that of frame directly.     

The Stone-Čech compactification of a partial frame via ideals and cozero elements

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 by S ; these partial frames are called S-frames.

We construct free frames over S-frames using appropriate ideals, called S-ideals. Taking S-ideals gives a functor from S-frames to frames. Coupled with the functor from frames to S-frames that takes S-Lindelöf elements, it provides a category equivalence between S-frames and a non-full subcategory of frames. In the setting of complete regularity, we provide the functor taking S-cozero elements which is right adjoint to the functor taking S-ideals. This adjunction restricts to an equivalence of the category of completely regular S-frames and a full subcategory of completely regular frames. As an application of the latter equivalence, we construct the Stone-? ech compactification of a completely regular S-frame, that is, its compact coreflection in the category of completely regular S-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.     

A chaotic function with a distributively scrambled set of full Lebesgue measure

Misiurewicz proved that there exists a continuous map of the interval [0, 1] onto itself for which there exists a scrambled set of full Lebesgue measure. In this paper, we form a continuous interval map which has a distributively scrambled set of full Lebesgue measure in which each point has dense orbit. This contains Misiurewicz’s result, since any distributively scrambled set must be scrambled but the converse is not generally true.     

Representation theorems for directed completions of consistent algebraic L-domains

In this paper, consistent algebraic L-domains are considered. One algebraic and two topological characterization theorems for their directed completions are given. It is proved that eliminating a set of maximal elements with empty interior from an algebraic L-domain results a consistent algebraic L-domain whose directed completion is just the given algebraic L-domain up to isomorphism. It is also proved that the category CALDOM of consistent algebraic L-domains and Scott continuous maps is Cartesian closed and has the category ALDOM of algebraic L-domains and Scott continuous maps as a full reflective subcategory. Received January 8, 2005; accepted in final form June 15, 2005.     

Conditions under which the least compactification of a regular continuous frame is perfect

We characterize those regular continuous frames for which the least compactification is a perfect compactification. Perfect compactifications are those compactifications of frames for which the right adjoint of the compactification map preserves disjoint binary joins. Essential to our characterization is the construction of the frame analog of the two-point compactification of a locally compact Hausdorff space, and the concept of remainder in a frame compactification. Indeed, one of the characterizations is that the remainder of the regular continuous frame in each of its compactifications is compact and connected.     

POWERS AND EXPONENTIAL OBJECTS IN INITIALLY STRUCTURED CATEGORIES AND APPLICATIONS TO CATEGORIES OF LIMIT SPACES

Abstract

Generalizing results of Herrlich and Nel, the author characterizes by means of smallest proper structures those objects X of an initially structured category for which X x—has a right adjoint, and describes the corresponding function spaces. It is shown that reduction to finally and initially dense classes is possible. The results are applied to epireflective subcategories of the category of limit spaces containing a finite non-indiscrete space, in particular to epireflective subcategories of TOP.     

Remarks on the frame envelope of a σ-frame

The familiar equivalence between σ-frames and σ-coherent frames, given by the frame envelopes of σ-frames, is shown to induce an equivalence between stably continuous σ-frames and stably continuous frames. Similarly, the analogue of the former for σ-biframes is proved to provide an equivalence between compact regular σ-biframes and compact regular biframes. As an application we obtain the equivalence between stably continuous σ-frames and compact regular σ-biframes due to Matutu as an easy consequence of its frame counterpart established earlier by Banaschewski and Brümmer. This provides an affirmative answer to a question posed by Dana Scott.     

Join-continuous frames,Priestley's duality and biframes

The primary purpose of this paper is to study join-continuous frames. We present two representation theorems for them: one in terms of -subframes of complete Boolean algebras and the other in terms of certain Priestley spaces. This second representation is used to prove that the topological spaces whose frame of open sets is join-continuous are characterized by a condition which says that certain intersections of open sets are open. Finally, we show that Priestley's duality can be viewed as a partialization of the dual adjunction between the categories of, respectively, bitopological spaces and biframes, stated by B. Banaschewski, G. C. L. Brümmer and K. A. Hardie in [5].This work was partially supported by Centro de Matemáíica da Universidade de Coimbra.     

On fiber products of elementary maps

The following characterization of fully closed maps is proved: a quotient map between regular spaces is fully closed if and only if it coincides with the fiber product of elementary maps between regular spaces.