On finite complete rewriting systems, finite derivation type, and automaticity for homogeneous monoids

TL;DR

This paper explores the computational properties of homogeneous monoids, demonstrating the coexistence of various properties and their negations, and introduces new concepts and techniques to extend these results.

Contribution

It introduces the concept of abstract Rees-commensurability and a new encoding technique to analyze homogeneous monoids' properties.

Findings

Existence of homogeneous monoids with all combinations of properties and negations.

Extension of results to n-ary homogeneous monoids.

Partial extension to n-ary multihomogeneous monoids.

Abstract

This paper investigates the class of finitely presented monoids defined by homogeneous (length-preserving) relations from a computational perspective. The properties of admitting a finite complete rewriting system, having finite derivation type, being automatic, and being biautomatic are investigated for this class of monoids. The first main result shows that for any consistent combination of these properties and their negations, there is a homogeneous monoid with exactly this combination of properties. We then introduce the new concept of abstract Rees-commensurability (an analogue of the notion of abstract commensurability for groups) in order to extend this result to show that the same statement holds even if one restricts attention to the class of $n$-ary homogeneous monoids (where every side of every relation has fixed length $n$). We then introduce a new encoding technique that allows us to extend the result partially to the class of $n$-ary multihomogenous monoids.