Semi-uninorms and implications on a complete lattice

An extension of a uninorm called a semi-uninorm is introduced and discussed in this paper. First, we introduce the concept of semi-uninorms on a complete lattice. Then, we discuss two kinds of residual operators of semi-uninorms and give conditions such that the operators are implications. We also give equivalent conditions for infinitely ∨∨-distributive left- and right-conjunctive semi-uninorms. Furthermore, we define two classes of induced operators by implications on a complete lattice and give conditions such that they are semi-uninorms. We also provide the equivalent conditions for the infinitely ∧∧-distributive implications in their second variables.

► Semi-uninorms are introduced and their two kinds of residual operators are discussed. ► Equivalent conditions for Ú-distributive conjunctive semi-uninorms are given. ► The conditions such that the induced operators by implications are semi-uninorms are given. ► Equivalent conditions for infinitely Ú-distributive implications are provided.