Rees coextensions of finite, negative tomonoids

TL;DR

This paper characterizes and provides a method to generate all Rees coextensions of finite, negative tomonoids, which are totally ordered monoids with the monoidal identity as the top element, relevant in fuzzy logic.

Contribution

It offers a complete characterization and a stepwise construction method for Rees coextensions of finite, negative tomonoids using level-set representations.

Findings

Characterization of Rees coextensions by one element larger

A stepwise method to generate all such tomonoids

Use of level-set representations to identify structures

Abstract

A totally ordered monoid, or tomonoid for short, is a monoid endowed with a compatible total order. We deal in this paper with tomonoids that are finite and negative, where negativity means that the monoidal identity is the top element. Examples can be found, for instance, in the context of finite-valued fuzzy logic. By a Rees coextension of a negative tomonoid $S$, we mean a negative tomonoid $T$ such that a Rees quotient of $T$ is isomorphic to $S$. We characterise the set of all those Rees coextensions of a finite, negative tomonoid that are by one element larger. We thereby define a method of generating all such tomonoids in a stepwise fashion. Our description relies on the level-set representation of tomonoids, which allows us to identify the structures in question with partitions of a certain type.