Constructing left-continuous triangular norms on complete lattices

TL;DR

This paper introduces new methods for constructing left-continuous t-norms on complete lattices using $rak{f}$-mappings and weak $rak{f}$-mappings, providing theoretical conditions and characterizations.

Contribution

It develops a novel framework using $rak{f}$-mappings to generate and characterize left-continuous t-norms on complete lattices, including conditions for ordinal sums.

Findings

Weak $rak{f}$-mappings induce left-continuous t-subnorms

$rak{f}$-mappings generate left-continuous t-norms under certain lattice conditions

Necessary and sufficient conditions for ordinal sum operators to be left-continuous t-norms

Abstract

This article focuses on the construction of left-continuous t-norms on complete lattices. The concepts of $\mathfrak{f}$-mappings and weak $\mathfrak{f}$-mappings on complete lattices are first introduced, respectively. They are then applied to establish the following key results: weak $\mathfrak{f}$-mappings are used to induce left-continuous t-subnorms; $\mathfrak{f}$-mappings are used to generate left-continuous t-norms whenever the top element $1$ of the complete lattice is a completely join-irreducible element. Finally, some necessary and sufficient conditions are provided for an operator constructed by the ordinal sum of a series of annihilating binary operators being a left-continuous t-norm on a complete lattice.