Rewriting Systems on Arbitrary Monoids

TL;DR

This paper introduces monoidal rewriting systems (MRS) as an abstraction of string rewriting over arbitrary monoids, providing categorical analysis, a biadjunction with monoids, and a classification of presentations via transformations.

Contribution

It develops the theory of monoidal rewriting systems, establishes a categorical framework, and classifies presentations using generalized transformations, extending rewriting theory beyond free monoids.

Findings

Defined the 2-category of Noetherian Confluent MRS

Proved a biadjunction between MRS and monoids

Classified monoid presentations up to GETT-equivalence

Abstract

In this paper, we introduce monoidal rewriting systems (MRS), an abstraction of string rewriting in which reductions are defined over an arbitrary ambient monoid rather than a free monoid of words. This shift is partly motivated by logic: the class of free monoids is not first-order axiomatizable, so "working in the free setting" cannot be treated internally when applying first-order methods to rewriting presentations. To analyze these systems categorically, we define $\mathbf{NCRS_2}$ as the 2-category of Noetherian Confluent MRS. We then prove the existence of a canonical biadjunction between $\mathbf{NCRS_2}$ and $\mathbf{Mon}$. Finally, we classify all Noetherian Confluent MRS that present a given fixed monoid. For this, we introduce Generalized Elementary Tietze Transformations (GETTs) and prove that any two presentations of a monoid are connected by a (possibly infinite) sequence of these transformations, yielding a complete characterization of generating systems up to GETT-equivalence.