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. Keywords: fuzzy relations, relation algebras, representation theorem, Dedekind categories, Zadeh category. 1 Introduction Since Zadeh's invention the concept of fuzzy sets has been extensively investigated in mathematics, science and engineering. The notion of fuzzy relations is also a basic one in processing fuzzy information in relational structures, see e.g. Pedrycz [10]. Goguen [2] generalized the concepts of fuzzy sets and relations taking values on partially ordered sets. Fuzzy relational equations were initiated and applied to medical models of diagnosis by Sanchez [12]. On the other hand theory of relations, namely relational calculus, has a long history, see [8, 13, 14] for...

This paper provides an algebraic characterization of mathematical structures formed by cartesian products of fuzzy relations with sup-min composition. A simple proof of a representation theorem for Boolean relation algebras satisfying Tarski rule and point axiom was given by G. Schmidt and T. Strohlein, and cartesian products of Boolean relation algebras were investigated by B. J'onsson and A. Tarski. Unlike Boolean relation algebras, fuzzy relation algebras are not Boolean but equipped with semi-scalar multiplication. First we present a set of axioms for fuzzy relation algebras and add axioms for cartesian products of fuzzy relation algebras. Second we improve the definition of point relations. Then a representation theorem for such relation algebras is deduced. Keywords : fuzzy relations, cartesian products, relation algebras, representation theorem. 1 Introduction In 1941 Tarski [8] proposed a problem, that is, "Is every relation algebra isomorphic to an algebra of all Bo...

This paper provides a proof of a representation theorem for homogeneous relation algebras by using concepts of scalar relations and point relations. Keywords : L-relations, relation algebras, scalar relations, point relations, representation theorem. 1 Introduction Just after Zadeh's work on fuzzy sets in 1965, Goguen [1] generalized the concepts of fuzzy sets and relations to taking values on arbitrary lattices, and also stressed the importance of relations as follows: The importance of relations is almost self-evident. Science is, in a sense, the discovery of relations between observables. Zadeh has shown the study of relations to be equivalent to the general study of systems (a system is a relation between an input space and an output space). The modern algebraic study of (binary) relations, namely relational calculus, was begun by Tarski; see [12] for details of the history of the study of Boolean relation algebras. In [10] Tarski proposed a formalisation of Boolean relation ...

The aim of this thesis is to develop the fuzzy relational calculus. To develop this calculus, we study four algebraic formalisations of fuzzy relations which are called fuzzy relation algebras, Zadeh categories, relation algebras and Dedekind categories, and we strive to arrive at their representation theorems. The calculus of relations has been investigated since the middle of the nineteenth century. The modern algebraic study of (binary) relations, namely relational calculus, was begun by Tarski. The categorical approach to relational calculus was initiated by Mac Lane and Puppe, and Dedekind categories were introduced by Olivier and Serrato. The representation problem for Boolean relation algebras was proposed by Tarski as the question whether every Boolean relation algebra is isomorphic to an algebra of ordinary homogeneous relations. There are many sufficient conditions that guarantee representability for Boolean relation algebras. Schmidt and Strohlein gave a simple proof of the...