Bivalent and other solutions of fuzzy relational equations via linguistic hedges

Abstract

Abstract We show that the well-known results regarding solutions of fuzzy relational equations and their systems can easily be generalized to obtain criteria regarding constrained solutions such as solutions which are crisp relations. When the constraint is empty, constrained solutions are ordinary solutions. The generalization is obtained by employing intensifying and relaxing linguistic hedges, conceived in this paper as certain unary functions on the scale of truth degrees. One aim of the paper is to highlight the problem of constrained solutions and to demonstrate that this problem naturally appears when identifying unknown relations. The other is to emphasize the role of linguistic hedges as constraints. © 2011 Elsevier B.V. All rights reserved. Motivation Fuzzy relational equations play an important role in fuzzy set theory and its applications, see and every fuzzy relation U satisfying the first or the second equality is called a solution of the respective fuzzy relational equation. The nature of the unknown relationship represented by U may impose additional constraints on U. For example, one may require that U be a bivalent (crisp) relation (see Section 3 for an illustrative example). More generally