Construction of 2-uninorms on bounded lattices

Document Type : Research Paper

Authors

1 Department of Mathematics, School of Science, Nanchang University, Nanchang, Jiangxi, 330031, China

2 College of Mathematics and Information Science, Nanchang Normal University, Nanchang, Jiangxi, 330031, China

Abstract

Uninorms and nullnorms are special 2-uninorms. In this work, we  construct 2-uninorms on bounded lattices. Let L be a bounded lattice with a nontrivial element d. Given two uninorms U_1 and U_2, defined on sublattices [0,d] and [d,1], respectively, this paper presents two methods for constructing binary operators on L which extend both U_1 and U_2.  We show that our first construction is a 2-uninorm on L if and only if U_2 is  conjunctive  and our second construction is a 2-uninorm on L if and only if U_1 is  disjunctive. Moreover, we prove that the two  2-uninorms are, respectively,  the  weakest  and the strongest 2-uninorm among all 2-uninorms, the restrictions of which on [0,d]^2 and [d,1]^2 are respectively U_1 and U_2.