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

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.     

Algebraic quantum kk-theory

Let 𝒢 be an algebraic quantum group. We introduce an equivariant algebraic kk-theory for 𝒢-module algebras. We study an adjointness theorem related with smash product and trivial action. We also discuss a duality property.