Algebraic Formalisations of Fuzzy Relations

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 su cient conditions that guarantee representability for Boolean relation algebras. Schmidt and Strohlein gave a simple proof of the representation theorem by using a notion of point relations. This notion is a relational (algebraic) characterisation of elements in a set. They also showed the important rôle of point relations in computer science. Following the spirit of Schmidt and Strohlein, we prove representation theorems for algebraic formalisations of fuzzy relations in this thesis.