Dualities and dual pairs in Heyting algebras

We extract the abstract core of finite homomorphism dualities using the techniques of Heyting algebras and (combinatorial) categories.Comment: 17 pages; v2: minor correction

On the Monadic Second-Order Transduction Hierarchy

We compare classes of finite relational structures via monadic second-order transductions. More precisely, we study the preorder where we set C \subseteq K if, and only if, there exists a transduction {\tau} such that C\subseteq{\tau}(K). If we only consider classes of incidence structures we can completely describe the resulting hierarchy. It is linear of order type {\omega}+3. Each level can be characterised in terms of a suitable variant of tree-width. Canonical representatives of the various levels are: the class of all trees of height n, for each n \in N, of all paths, of all trees, and of all grids

HOMOMORPHISMS OF STRUCTURES (CONCEPTS AND HIGHLIGHTS)

Abstract. In this paper we survey the recent results on graph homomorphisms perhaps for the first time in the broad range of their relationship to wide range applications in computer science, physics and other branches of mathematics. We illustrate this development in each area by few results. 1

Towards Mathematical Aesthetics \Lambda

Abstract We discuss a possibility of visual data global analysis with respect to the aesthetic (harmonious) criteria. We explain our approach that is mathematical based and expressed by the Hereditary Entropy Thesis. We give some computational evidence which outlines aims and indicates meaning of this project

Universality of separoids

summary:A separoid is a symmetric relation \dagger \subset {2^S\atopwithdelims ()2} defined on disjoint pairs of subsets of a given set $S$ such that it is closed as a filter in the canonical partial order induced by the inclusion (i.e., $A\dagger B\preceq A^{\prime }\dagger B^{\prime }\iff A\subseteq A^{\prime }$ and $B\subseteq B^{\prime }$). We introduce the notion of homomorphism as a map which preserve the so-called “minimal Radon partitions” and show that separoids, endowed with these maps, admits an embedding from the category of all finite graphs. This proves that separoids constitute a countable universal partial order. Furthermore, by embedding also all hypergraphs (all set systems) into such a category, we prove a “stronger” universality property. We further study some structural aspects of the category of separoids. We completely solve the density problem for (all) separoids as well as for separoids of points. We also generalise the classic Radon’s theorem in a categorical setting as well as Hedetniemi’s product conjecture (which can be proved for oriented matroids)

A NEW COMBINATORIAL APPROACH TO THE CONSTRAINT SATISFACTION PROBLEM DICHOTOMY CLASSIFICATION

Abstract. We introduce a new general polynomial-time construction- the fibre construction- which reduces any constraint satisfaction problem CSP(H) to the constraint satisfaction problem CSP(P), where P is any subprojective relational structure. As a consequence we get a new proof (not using universal algebra) that CSP(P) is NP-complete for any subprojective (and so for any projective) relational structure. This provides a starting point for a new combinatorial approach to the NP-completeness part of the conjectured Dichotomy Classification of CSPs, which was previously obtained by algebraic methods. This approach is flexible enough to yield solution of two problems related to NP-completeness of coloring graphs with large girth and bounded degree restrictions. 1

Gaps And Dualities In Heyting Categories

We present an algebraic treatment of the correspondence of gaps and dualities in partial ordered classes induced by the morphism structures of certain categories which we call Heyting (such are for instance all cartesian closed categories, but we present examples of others, too). This allows to extend the results of [12] to a wide range of more general structures. Also, we introduce a notion of combined dualities and discuss the relation of their structure to that of the plain ones