PROCESSING IN RELATIONAL STRUCTURES - FUZZY RELATIONAL EQUATIONS

The paper provides an overview of methodology for processing fuzzy information in relational structures completed in terms of fuzzy relational equations. The origin and the central role of relational calculus in general, and fuzzy sets in particular, is explained with special emphasis paid to its role and representation capabilities. It is pointed out that fuzzy relational equations play a significant role as a platform for a uniform development of techniques in fuzzy sets. Many of the problems and frameworks developed in fuzzy sets so far can be easily translated into the language of fuzzy relational equations which immediately takes a significant advantage of their solid, well developed formalisms, techniques, and clarity of exposition. We will study the role of approximate solutions to fuzzy relational equations as a convenient tool to handle probabilistic type of uncertainty. Moreover those solutions can easily explain the origin and generate a way in which fuzzy sets of higher generality (such as for instance type-2 sets, interval-valued sets, . . .) can be algorithmically determined. Finally, we study the role of equations in diverse fields of applications making use of an ample framework of general systems theory.