A survey on fuzzy relational equations, part I: classification and solvability

Abstract

Fuzzy relational equations play an important role in fuzzy set theory and fuzzy logic systems, from both of the theoretical and practical viewpoints. The notion of fuzzy relational equations is associated with the concept of "composition of binary relations." In this survey paper, fuzzy relational equations are studied in a general lattice-theoretic framework and classified into two basic categories according to the duality between the involved composite operations. Necessary and sufficient conditions for the solvability of fuzzy relational equations are discussed and solution sets are characterized by means of a root or crown system under some specific assumptions.

On the resolution of bipolar max-min equations

“…The system A − • ¬x = b can be handled analogously and its solution set, if not empty, can be characterized by a minimum solution and finitely many maximal solutions. For some detailed discussion on fuzzy relational equations, see, e. g., Di Nola et al [3], De Baets [2], Peeva and Kyosev [11], Li and Fang [7,8], Li [6], and references therein.…”

Obtaining representable coimplications from aggregation and dual operators

“…7 states how an implication gives rise to a coimplication and vice-versa, see also [30,31,38] and [4].…”

Linear optimization with bipolar max-parametric hamacher fuzzy relation equation constraints

“…A comprehensive review about analytical approaches of FRE resolution can be found in [4,8]. Good overviews about theory and applications of FREs can also be found in [6,12,13,16,18,26,27,28]. The problem of minimizing a linear function with max-min FRE constraints was first investigated by Fang and Li [9] in 1999.…”