Bivalent and other solutions of fuzzy relational equations via linguistic hedges

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.

► We study constrained solutions of fuzzy relational equations and their systems. ► Examples of constrained solutions are bivalent (crisp) solutions. ►Constrained solutions are found as ordinary solutions of generalized equations. ►The generalization is based on employing linguistic hedges. ► We provide the criteria of solvability of generalized equations and their systems.