Triangular norms on bounded trellises

TL;DR

This paper explores the concept of t-norms on bounded trellises, highlighting the effects of non-transitivity, and introduces a generic construction method for extending t-norms within these structures.

Contribution

It introduces the notion of t-norms on bounded trellises, analyzes the impact of non-transitivity, and proposes a generic extension method for t-norms in these settings.

Findings

Meet operation is not a t-norm on proper bounded trellises.

Multiple maximal t-norms can exist on a bounded trellis.

A generic construction method for extending t-norms is proposed.

Abstract

In this paper, we introduce the notion of a t-norm on bounded pseudo-ordered sets and in particular on bounded trellises (also known as weakly associative lattices), and provide some basic examples. The impact of abandoning transitivity is considerable: on a proper bounded trellis, the meet operation is not a t-norm, and there might actually exist no or even multiple maximal t-norms. We provide a first generic construction method that allows to extend a t-norm on an interior range of a given $\meet$-semi-trellis to the entire $\meet$-semi-trellis. Also, we discuss at length an instantiation of this method based on a particular interior range, namely a finite sub-trellis of the set of right-transitive elements of a given trellis. We pay specific attention to bounded pseudo-chains and modular trellises.