Real coextensions as a tool for constructing triangular norms

TL;DR

This paper introduces a universal method for constructing left-continuous triangular norms using extensions of ordered monoids, providing a systematic way to generate such norms with explicit construction details and illustrative examples.

Contribution

The paper presents a novel, systematic method for constructing left-continuous t-norms from ordered monoids, expanding the toolkit for fuzzy logic and related fields.

Findings

Method successfully constructs l.-c. t-norms from ordered monoids.

Explicit conditions and constituents for the construction are provided.

Several illustrative examples demonstrate the method's applicability.

Abstract

We present in this paper a universal method of constructing left-continuous triangular norms (l.-c. t-norms). The starting point is an arbitrary, possibly finite, totally ordered monoid fulfilling the conditions that are characteristic for l.-c. t-norms: commutativity, negativity, and quanticity. We show that, under suitable conditions, we can extend this structure by substituting each element for a real interval. The process can be iterated and if the final structure obtained in this way is order-isomorphic to a closed real interval, its monoidal operation can, up to isomorphism, be identified with a l.-c. t-norm. We specify the constituents needed for the construction in an explicit way. We furthermore illustrate the method on the basis of a number of examples.