Dedekind Categories Research Articles

Using relation-algebraic means and tool support for investigating and computing bipartitions

Using Dedekind categories as an algebraic structure for (binary) set-theoretic relations without complements, we present purely algebraic definitions of “to be bipartite” and “to possess no odd cycles” and prove that both notions coincide in case that the reflexive–transitive closure of the relation in question is symmetric. This generalises D. Kőnig's well-known theorem from undirected graphs to abstract relations and, hence, to models such as L-relations that are different from set-theoretic relations. One direction of this generalisation holds for general relations. The other direction requires the symmetry of the reflexive–transitive closure as premise and is shown by specifying a bipartition for the given relation in form of a pair of disjoint relational vectors. For set-theoretic relations this immediately leads to a relational program for computing a bipartition. We also investigate the structure of a certain set of bipartitions that is generated by a given bipartition, explore the set of all bipartitions of a bipartite relation with respect to minimality and maximality of elements in view of the component-wise inclusion, show a relationship between bipartitions and bichromatic partitions and also discuss how the algebraic proofs can be mechanised using theorem proving tools.

Dependencies in relational models of databases

In order to provide a common framework for classical, fuzzy and L-fuzzy databases we assume that databases are set-valued, i.e., that every value of an attribute is a set in some suitable set theory. Tables of a database are modeled using the theory of Dedekind categories and its extensions. These theories establish an abstract approach to binary, classical (crisp) or fuzzy relations. Based on this model we investigate dependencies in databases in a general setting.

Categories of relations for variable-basis fuzziness

Arrow categories establish a categorical and algebraic description of L-fuzzy relations, i.e., relations that use membership values from an arbitrary but fixed complete Heyting algebra L. With other words arrow categories describe the fixed-basis case. In this paper we are interested in the variable-basis case, i.e., the case where relations between different objects may use different membership values. We will investigate the structure of the collection of lattices of membership values within a given Dedekind category. This will lead to a complete characterization of the variable-basis case in this context.

Point axioms and related conditions in Dedekind categories

A Dedekind category is a convenient algebraic framework to manipulate (binary) relations. Concepts of points, point axioms and related conditions such as the axiom of totality, the axiom of subobject, the axiom of complement, and the relational axiom of choice are introduced in Dedekind categories in order to connect abstract notions to set-theoretical intuition. This paper summarises logical interrelations of these axioms and provides some ideas for using them.

Relational properties of sequential composition of coalgebras

In this paper we define a sequential composition for arbitrary coalgebras in a Dedekind category. We show some basic algebraic properties of this operation up to bisimulation. Furthermore, we consider certain recursive equations and provide an explicit solution, i.e., a solution not based on an iterative process.

Dedekind categories with cutoff operators

Dedekind categories provide a suitable categorical framework for lattice-valued binary relations. It is known that the notion of crispness cannot be described by the basic tools of this theory only. In this paper we will study Dedekind categories with a cutoff operator in order to circumvent this shortage. We will introduce and investigate the properties of the operator and its relationship with other tools previously used for the same purpose. The main result of this paper is a representation theorem for Dedekind categories with a cutoff operator satisfying the point axiom.

TARSKI’S FIXED POINT THEOREM IN DEDEKIND CATEGORIES

Tarski’s fixed point theorem for complete lattices is a fundamental theorem in lattice theory and is very useful for computer science applications. This paper will show a general result in Dedekind categories, which extends the Tarski’s theorem.

Representation theory of Goguen categories

Goguen categories constitute a suitable algebraic formalisation for L -fuzzy relations. It is well-known that an L -fuzzy relation may be represented by the set of all its α-cuts. The aim of this paper is to show a similar result for Goguen categories. Furthermore, given an algebraic structure of relations, a Dedekind category R , and a complete Brouwerian lattice L , the idea above allows us to define a Goguen category G such that the underlying structures are R and L . Using our pseudo-representation theorem we show that the representation theory of Goguen categories is equivalent to the representation theory of simple Dedekind categories. This result allows us to transfer known representation results for Dedekind categories to the theory of Goguen categories.

A new algebraic approach to L-fuzzy relations convenient to study crispness

Abstract The aim of this paper is to develop a suitable calculus of L -fuzzy relations. We show that for this purpose the theory of Dedekind categories is too weak. Therefore, we introduce the notion of a Goguen category as a suitable algebraic definition and show several properties of this kind of a relational category.

Categorical representation theorems of fuzzy relations

This paper provides a notion of Zadeh categories as a categorical structure formed by fuzzy relations with sup-min composition, and proves two representation theorems for Dedekind categories (relation categories) with a unit object analogous to one-point set, and for Zadeh categories without unit objects.

Squares and rectangles in relation categories Three cases: semilattice, distributive lattice and boolean non-unitary

In this paper we investigate rectangle morphisms of Dedekind categories (or semilattice relation categories, or unitary allegories) which appear to be a basic structure for relation categories. In order to reconstruct the Dedekind category generated by all the rectangle morphisms, we are obliged to clarify the role of both the square morphisms and the functions s which assign to every square morphism c the ideal morphism cs = 1 c1. We translate this observations into adapted new objects that we call the “MP-families”. In this way, the reconstruction problem becomes an adjunction problem. To each Dedekind category we associate a MP-family; to each functor between two Dedekind categories we associate a morphism between two MP-families. This association is a functor. This functor has a left adjoint. This result is also true for other universes: distributive lattices, Boolean algebras.