Ordinal sum of two binary operations being a t-norm on bounded lattice

TL;DR

This paper establishes necessary and sufficient conditions under which the ordinal sum of two binary operations on a bounded lattice results in a t-norm, addressing an open problem in the field.

Contribution

It provides new criteria ensuring the ordinal sum of binary operations on bounded lattices is a t-norm, extending previous understanding and solving an open problem.

Findings

Identifies conditions for ordinal sums to be t-norms

Provides a solution to an open problem in lattice theory

Enhances methods for constructing t-norms on bounded lattices

Abstract

The ordinal sum of t-norms on a bounded lattice has been used to construct other t-norms. However, an ordinal sum of binary operations (not necessarily t-norms) defined on the fixed subintervals of a bounded lattice may not be a t-norm. Some necessary and sufficient conditions are presented in this paper for ensuring that an ordinal sum on a bounded lattice of two binary operations is, in fact, a t-norm. In particular, the results presented here provide an answer to an open problem put forward by Ertu\u{g}rul and Ye\c{s}ilyurt [Ordinal sums of triangular norms on bounded lattices, Inf. Sci., 517 (2020) 198-216].