A unified framework of fuzzy implications and coimplications

Fuzzy implications and coimplications play important roles in both theoretic and applied communities of fuzzy set theory. In this paper, we provide a unified framework for fuzzy implications and coimplications. Specifically, firstly, we introduce the concept of uni-implications, which is the unification of fuzzy implications and coimplications, and describe the structure of uni-implications. Secondly, we discuss the relationships among uni-implications, fuzzy boundary weak implications and fuzzy (co)implications. Thirdly, we present two constructions and equivalent characterizations of non-trivial uni-implications, respectively. Finally, we propose two binary operations ♠,♡ on some subset of the set of non-trivial uni-implications, denoted by FSNT, which make (FSNT,♠) a semigroup and (FSNT,♡) a non-commutative and non-idempotent monoid.