Intrinsic approach spaces on domains

The paper is a contribution to quantifiability of domains. We show that every domain X, regardless of cardinality conditions for a domain bases, is quantifiable in the sense that there exists an approach structure on X (Lowen (1997) [9]), defined by means of a gauge of quasi metrics, inducing the Scott topology. We get weightability for free and in the case of an algebraic domain satisfying the Lawson condition (Lawson (1997) [8]), a quantifying approach space can be obtained with a weight satisfying the kernel condition.     

Separation properties in algebraic categories of topological spaces

Full subcategories C ? Top of the category of topological spaces, which are algebraic over Set in the sense of Herrlich [2], have pleasant separation properties, mostly subject to additional closedness assumptions. For instance, every C-object is a T₁-space, if the two-element discrete space belongs to C. Moreover, if C is closed under the formation of finite powers in Top and even varietal [2], then every C-object is Hausdorff. Hence, the T₂-axiom turns out to be (nearly) superfluous in Herrlich's and Strecker's characterization of the category of compact Hausdorff spaces [1], although it is essential for the proof.If we think of C-objects X as universal algebras (with possibly infinite operations), then the subalgebras of X form the closed sets of a compact topology on X, provided that the ordinal spaces [0, β] belong to C. This generalizes a result in [3]. The subalgebra topology is used to prove criterions for the Hausdorffness of every space in C, if C is only algebraic.     

Hausdorff separation in categories

Considering subobjects, points and a closure operator in an abstract category, we introduce a generalization of the Hausdorff separation axiom for topological spaces: the notion ofT ₂-object. We discuss the properties ofT ₂-objects, which depend essentially on the behaviour of points, and finally we relate them to the well-known separated objects.The results of this paper are essentially taken from the author's Ph. D. Thesis written under the supervision of Professors M. Sobral and W. Tholen and partially supported by a scholarship of I.N.I.C.-Instituto Nacional de Investigação Científica.     

Closed structures on categories of topological spaces

The category of all topological spaces and continuous maps and its full subcategory of all T_o-spaces admit (up to isomorphism) precisely one structure of symmetric monoidal closed category (see [2]). In this paper we shall prove the same result for any epireflective subcategory of the category of topological spaces (particularly e.g. for the categories of Hausdorff spaces, regular spaces, Tychonoff spaces).     

Ordered topological structures

The paper discusses interactions between order and topology on a given set which do not presuppose any separation conditions for either of the two structures, but which lead to the existing notions established by Nachbin in more special situations. We pursue this discussion at the much more general level of lax algebras, so that our categories do not concern just ordered topological spaces, but also sets with two interacting orders, approach spaces with an additional metric, etc.     

Finitely generated varieties of distributive double <Emphasis Type="Italic">p</Emphasis>-algebras universal modulo a group

For any monoid M, any universal variety contains arbitrarily large algebras whose endomorphism monoid is isomorphic to M. A variety universal modulo a group G contains arbitrarily large algebras whose endomorphism monoid is isomorphic to the direct product M x G. One of the results of this paper structurally characterizes all finitely generated varieties of distributive double p-algebras universal modulo a group, and shows that any unavoidable direct factor G is a Boolean group with at most eight elements.     

Value semigroups,value quantales,and positivity domains

In 1981 and 1997 Kopperman and Flagg, respectively, proved that every topological space is metrisable, provided the symmetry and separation axioms are removed from the requirements on the metric, and the metric is allowed to take values in, respectively, a value semigroup or a value quantale. Seeking to construct a value quantale from a value semigroup we focus on a small portion of the structure present in a value semigroup, comprising what we call a positivity domain, and we construct its enveloping value quantale, forming part of a detailed comparison between value semigroups and value quantales. We obtain a representation theorem for value quantales in terms of positivity domains, and we outline how products of positivity domains can be used in the theory of continuity spaces instead of (the non-existent) products of value quantales.     

Approximation in quantale-enriched categories

Our work is a foundational study of the notion of approximation in Q-categories and in (U,Q)-categories, for a quantale Q and the ultrafilter monad U. We introduce auxiliary, approximating and Scott-continuous distributors, the way-below distributor, and continuity of Q- and (U,Q)-categories. We fully characterize continuous Q-categories (resp. (U,Q)-categories) among all cocomplete Q-categories (resp. (U,Q)-categories) in the same ways as continuous domains are characterized among all dcpos. By varying the choice of the quantale Q and the notion of ideals, and by further allowing the ultrafilter monad to act on the quantale, we obtain a flexible theory of continuity that applies to partial orders and to metric and topological spaces. We demonstrate on examples that our theory unifies some major approaches to quantitative domain theory.     

Projective sup-algebras: a general view

We begin with the notion of K-flat projectivity. For each sup-algebra L we then introduce a binary relation L_? on it. The K-flat projective sup-algebras are exactly such sup-algebras with each element a approximated by the element x, xL_?a and the relation L_? being stable with respect to the operations on L. Further on, we introduce the notion of a K-comonad and characterize K-flat projective sup-algebras as such sup-algebras having a coalgebra structure for the K-comonad.     

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.     

Towards Stone duality for topological theories

In the context of categorical topology, more precisely that of T-categories (Hofmann, 2007 [8]), we define the notion of T-colimit as a particular colimit in a V-category. A complete and cocomplete V-category in which limits distribute over T-colimits, is to be thought of as the generalisation of a (co-)frame to this categorical level. We explain some ideas on a T-categorical version of “Stone duality”, and show that Cauchy completeness of a T-category is precisely its sobriety.     

Nuclear Typing of Frames vs Spatial Selectors

Nuclei which are defined over a class of frames are called nuclear typings. There is the dual notion of a spatial selector, and the relationship between nuclear typings and spatial selectors emanates from the duality between spatial frames and sober spaces. Especially interesting is the interplay between typings that are well-behaved with respect to certain frame quotients and selectors which similarly behave well in passage to closed sets.     

Geometric topological completions with universal final lifts

In this paper necessary and sufficient conditions are given on a concrete category over a category B so that it can be densely embedded (over B) into a geometric topological category E that admits certain universal final lifts. These conditions, as well as the class of universal final lifts, depend upon an a priori given full subcategory Δ of B. For example, E may have, depending upon Δ and B, universal coproducts or quotients or colimits. For appropriate Δ's, if B is cartesian closed then so is E.     

Initial morphisms and monomorphisms

Initial cones are shown to be mono-cones for several important types of functors, namely those possessing a reducible generator or having a certain topological (1.7) resp. algebraic property (1.8). These results are useful for the computation of the universal initial completion of faithful, topologically algebraic functors, contain results of Börger-Tholen and Hoffmann for certain setbased functors as special cases and allow a complete characterization of the initial cones in quite a few categories.     

Prime ends for domains in metric spaces

In this paper we propose a new definition of prime ends for domains in metric spaces under rather general assumptions. We compare our prime ends to those of Carathéodory and Näkki. Modulus ends and prime ends, defined by means of the p$p$-modulus of curve families, are also discussed and related to the prime ends. We provide characterizations of singleton prime ends and relate them to the notion of accessibility of boundary points, and introduce a topology on the prime end boundary. We also study relations between the prime end boundary and the Mazurkiewicz boundary. Generalizing the notion of John domains, we introduce almost John domains, and we investigate prime ends in the settings of John domains, almost John domains and domains which are finitely connected at the boundary.     

Exponential objects and Cartesian closedness in the constructPrtop

We give an internal characterization of the exponential objects in the constructPrtop and investigate Cartesian closedness for coreflective or topological full subconstructs ofPrtop. If $ is the set {0} {1/n;n 1} endowed with the topology induced by the real line, we show that there is no full coreflective subconstruct ofPrtop containing $ and which is Cartesian closed. With regard to topological full subconstructs ofPrtop we give an example of a Cartesian closed one that is large enough to contain all topological Fréchet spaces and allT ₁ pretopological Fréchet spaces.Aspirant NFWO     

Regularity in algebraic frames

This article considers algebraic frames in which the meet of two compact elements is compact, and, in that context, when the subframe of all regular elements is itself regular. Motivated by the study of a frame of convex ?-subgroups of a lattice-ordered group, a number of relevant sufficient conditions are given for this subframe to be regular. An example is given of a frame of convex ?-subgroups for which the subframe of regular elements is not regular.