Uninorms via two comparable closure operators on bounded lattices

TL;DR

This paper introduces new methods for constructing uninorms on bounded lattices using two comparable closure or interior operators, expanding the theoretical framework and including degenerate cases related to existing results.

Contribution

The paper develops novel construction techniques for uninorms on bounded lattices using pairs of comparable closure or interior operators, with conditions and special cases analyzed.

Findings

Construction methods depend on lattice structure and chosen operators.

Results include degenerate cases with single operators.

Some cases recover known uninorm results.

Abstract

In this paper, we propose novel methods for constructing uninorms using two comparable closure operators or, alternatively, two comparable interior operators on bounded lattices. These methods are developed under the necessary and sufficient conditions imposed on these operators. Specifically, the construction of uninorms for $(x ,y )\in ]0 ,e [\times]e ,1 [ \cup ]e ,1 [\times]0 ,e [$ depends not only on the structure of the bounded lattices but also on the chosen closure operators (or interior operators). Consequently, the resulting uninorms do not necessarily belong to $\mathcal{U}_{min}^{*}\cup \mathcal{U}_{min}^{1}$ (or $\mathcal{U}_{max}^{*}\cup\mathcal{U}_{max}^{0}$). Moreover, we present the degenerate cases of the aforementioned results, which are constructed using only a single closure operator or a single interior operator. Some of these cases correspond to well-known results documented in the literature.