Triangular norms on finite lattices

TL;DR

This paper characterizes triangular norms on finite lattices, providing methods for their generation, and explores their properties, including continuity and representation, on atomistic and Boolean lattices.

Contribution

It introduces a method to generate triangular norms on atomistic lattices and characterizes their continuity and representation on Boolean and non-Boolean lattices.

Findings

Every triangular norm on a non-Boolean atomistic lattice is not left-continuous.

$T_M$ is the uniquely left-continuous triangular norm on an atomistic Boolean lattice.

Each atomistic Boolean lattice can be represented by a family of triangular norms.

Abstract

This article intends to characterize triangular norms on a finite lattice. We first give a method for generating a triangular norm on an atomistic lattice by the values of atoms. Then we prove that every triangular norm on a non-Boolean atomistic lattice is not left-continuous and $T_M$ is the uniquely left-continuous triangular norm on an atomistic Boolean lattice. Furthermore, we show that each atomistic Boolean lattice can be represented by a family of triangular norms on an atomistic lattice with the same number of atoms. Finally, we construct a triangular norm on a finite lattice by restricting a triangular norm on an extended atomistic lattice of the finite lattice to the finite lattice.