Left-continuous pseudo-t-norms on modular lattices

TL;DR

This paper investigates the conditions under which left-continuous pseudo-t-norms exist on modular lattices, providing characterizations and structural insights into their properties and limitations.

Contribution

It establishes a necessary and sufficient condition for left-continuous t-norms on lattice sums and characterizes lattices supporting pseudo-t-norms using 1-distributivity.

Findings

Condition for existence of left-continuous t-norms on lattice sums

Characterization of complete atomistic lattices with pseudo-t-norms

Identification of forbidden structures in finite modular lattices

Abstract

This article focuses on the relationship between pseudo-t-norms and the structure of lattices. First, we establish a necessary and sufficient condition for the existence of a left-continuous t-norm on the ordinal sum of two disjoint complete lattices. Then, we define the $1$-distributivity of a lattice, which is applied for characterizing a complete atomistic lattice that has a left-continuous pseudo-t-norm. We also describe the forbidden structures of a finite modular lattice that is a $1$-distributive lattice, which is used for representing a kind of finite planar modular lattices that have left-continuous pseudo-t-norms.