Aleš Pultr

Frames and Locales

Preface.- Introduction.- I. Spaces and lattices of open sets.- II. Frames and locales. Spectra.- III. Sublocales.- IV. Structure of localic morphisms. The categories Loc and Frm.- V. Separation axioms.- VI. More on sublocales.-VII. Compactness and local compactness.- VIII. (Symmetric) uniformity and nearness.- IX. Paracompactness.- X. More about completion.- XI. Metric frames.- XII. Entourages, non-symmetric uniformity.- XIII. Connectedness.- XIV. The frame of reals and real functions.- XV. Localic groups.- Appendix I: Posets.- Appendix II: Categories.- Bibliography.- Index of Notation.- Index.

On classes of relations and graphs determined by subobjects and factorobjects

Abstract The classes of relations and graphs determined by subobjects and factorobjects are studied. We investigate whether such classes are closed under products, whether they are finitely generated by products and subobjects and whether a class can be described alternatively by subobjects and factorobjects. This is related to good characterizations.

CAUCHY POINTS OF UNIFORM AND NEARNESS FRAMES

Abstract The notion of Cauchy point (= regular Cauchy filter) and the corresponding Cauchy spectrum, for a nearness frame (= uniform without the star-refinement condition) are investigated in various directions, including basic motivation, several functorial aspects, and the recognition of the Cauchy spectrum as the ordinary spectrum of the completion, after the unique existence of the latter is obtained as a central new result in this context.

On qualitatively independent partitions and related problems

Abstract The paper is concerned with several related combinatorial problems one of which is that of estimating the numbers of qualitatively independent p -partitions. Besides nonconstructive basic estimates, a constructive procedure yielding not much worse ones is presented. In conclusion, some applications are shown.

Lattice-Valued Frames, Functor Categories, And Classes Of Sober Spaces

This chapter introduces lattice-valued frames or L- frames, related to traditional frames analogously to how L-topological spaces relate to traditional spaces, via level sets and level mappings viewed as systems of frame morphisms (Proposition 3.3.2). En route, the well-known S2 and LS2 functors [25, 10, 11, 12, 16, 17, 18, 29, 33, 43, 44, 48, 54, 55, 56, 57, 58, 61] relating traditional spaces and L-spaces, respectively, to their associated (semi)locales of open and L-open sets are analogized and modified. This study both gives new descriptions of classes of sober spaces extant in the literature and creates a new class of sober spaces, justifying examples for which are given in Chapter 17 [42] of this Volume.

Variants of openness

Algebraic conditions on frame homomorphisms representing various types of openness requirements on continuous maps are investigated. It turns out that several of these can be expressed in terms of formulas involving pseudocomplements. A full classification of the latter is presented which shows that they group into five equivalence classes and establishes the logical connections between them. Among the relation of our algebraic conditions to continuous maps between topological spaces, we establish that the coincidence of the algebraic and topological notion of openness is equivalent to the separation axiomTD for the domain space.

On a product dimension of graphs

The Dushnik-Miller dimension of a partially ordered set (X, <) is defined (see [ 11) as the least number of linear orderings L, ,..., L, of X such that <= nj L,. Equivalently, it is the least number of linearly ordered (Xi, Li) such that (X, <) can be embedded as a spanned subobject into X (Xi, Li) (see [lo]). In a more general setting, we are often encountered with the following situation: We are given a class q of objects of a given nature (a concrete category)-e.g., posets, graphs, digraphs, particular kinds of these, etc.-and a subclass 9 of g such that every C E g can be embedded into a Xi”= 1 Bi with Bi E 9. It is then natural to regard the necessary number of the Bi as a measure of complexity of C, called the dimension (with respect to 59 and 5Y) of c. There is something like a “most canonical” 9 associated with a given V, namely, the class of all subdirectly irreducible objects (to avoid difficulties, let us assume the g in question to consist of finite objects). This is the case with the Dushnik-Miller dimension above if we regard the partial orderings as antireflexive, the linear orderings being the subdirectly irreducibles in the category in question. This is also the case with the dimension of graphs (symmetric graphs without loops) we discuss in this paper. Another example is the dim, studied, e.g., in [ 141: if we consider the category of reflexive partial orderings, the complete system of subdirectly irreducibles consists just of the l-chain and the 2-chain. Similarly, one could consider a dimension of a bipartite graph X as the least n such that X is isomorphic to a subobject of Pi, where P, is the 3-path, more generally, a dimension of a k-chromatic graph as the least n such that X is isomorphic to a subgraph of (P3 @ KkvZ)*, etc. (see [8, 121; for related representations of graphs see also (5, 131). Sometimes, a larger 9 may be more convenient. There are cases where there are many objects which, for given purposes, should be considered as basically simple although they are reducible. For example, representing

Examples For Different Sobrieties In Fixed-Basis Topology

In a previous chapter [21], the authors introduced a new approach to describing L-topological spaces using categorical constructs called fuzzy frames. This approach not only gives new decriptions of previously known types of sober spaces, but it also leads naturally to a new type of sober spaces not previously documented in the literature.

Productive classes and subdirect irreducibility, in particular for graphs

Abstract Given graphs A1,…,An (more generally, objects of a category ) denote by (A1,…,An) ¬ Graph (more generally, (A1,…,An) ) the class of all the graphs which contain no A, as a full subgraph (subobject in some sense). The question when (A1,…,An) is closed under products is studied. A is productive iff A is subdirectly irreducible. A characterization of subdirectly irreducibles enabling us to list them explicitely in some concrete cases is given, (A1,…An) is productive iff, in a suitable order, each A, is subdirectly irreducible in (A1,…,An) . The couples (A1, A2) for which (A1, A2)¬Graph is productive are listed.

Colored graphs without colorful cycles

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 mon ≡ 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.