Structures of uninorms on bounded lattices via t-conorms, closure operators and t-subnorms

TL;DR

This paper characterizes the structure of uninorms on bounded lattices using t-conorms, closure operators, and t-subnorms, establishing necessary and sufficient conditions and exploring dual constructions with t-norms and related operators.

Contribution

It introduces a comprehensive framework linking uninorms on bounded lattices with t-conorms, closure operators, and t-subnorms, including degenerate and dual cases, with proven necessary and sufficient conditions.

Findings

Established necessary and sufficient conditions for uninorm structures.

Presented dual constructions using t-norms, interior operators, and t-superconorms.

Connected uninorms with well-known results through special cases.

Abstract

In this paper, we provide some structures of uninorms on bounded lattices via t-conorms, closure operators and t-subnorms, subject to certain constraints on the closure operators and t-subnorms. Importantly, these constraints are shown to be both sufficient and necessary. That is, the proposed methods reveal clear relationships between the structure of the resulting uninorms and the properties of the underlying components. Meanwhile, we present the degenerate cases of the aforementioned results, which are constructed using special closure operators and t-subnorms. Some of these cases correspond to well-known results documented in the literature. Moreover, the dual constructions of uninorms on bounded lattices, based on t-norms, interior operators and t-superconorms, are presented simultaneously.