A new perspective on constructing 2-uninorms on bounded lattices

Abstract

Recently, Ertu\u{g}rul provided a way to obtain a 2-uninorm on a bounded lattice $L$ by using a disjunctive uninorm and a conjunctive uninorm. Later, Xie and Yi proposed two methods for constructing 2-uninorms on $L$ by using two uninorms $U_{1}$ on $[0_{L},k]$ and $U_{2}$ on $[k,1_{L}]$, and showed that the function constructed by the first method is a 2-uninorm iff $U_{2}$ is conjunctive and the function constructed by the second method is a 2-uninorm iff $U_{1}$ is disjunctive. Motivated by the three methods, we present two approaches to construct a 2-uninorm on $L$ via a uni-nullnorm and a t-conorm (a t-norm and a null-uninorm).
By the first new one, we can obtain a 2-uninorm on $L$ such that the uninorm on $[0_{L},k]$ is not necessarily disjunctive and the uninorm on $[k,1_{L}]$ is not necessarily conjunctive. The second approach is different from all existing construction ways for 2-uninorms on $L$.