Characterizing idempotent nullnorms on bounded lattices

TL;DR

This paper characterizes the conditions for the existence of idempotent nullnorms with a zero element on bounded lattices, expanding understanding of their structure and properties.

Contribution

It provides an equivalent characterization for idempotent nullnorms with a zero element on bounded lattices with incomparable elements, and explores related lattice properties.

Findings

Characterization of nullnorm existence conditions

Properties of bounded lattices with incomparable elements

Extension of nullnorm theory to more complex lattice structures

Abstract

Nullnorms with a zero element being at any point of a bounded lattice are an important generalization of triangular norms and triangular conorms. This paper obtains an equivalent characterization for the existence of idempotent nullnorms with the zero element $a$ on any bounded lattice containing two distinct elements incomparable with $a$. Furthermore, some basic properties for the bounded lattice containing two distinct element incomparable with a are presented.