Rewriting Systems on Arbitrary Monoids

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 2-adjunction between $\mathbf{NCRS_2}$ and $\mathbf{Mon}$. Finally, we study all MRS that present a given fixed monoid $M$. For this, we introduce Generalized Elementary Tietze Transformations (GETTs) and, among other results, we prove the property that any presentation of $M$ can be obtained from any other using GETTs is equivalent to $M$ being a free monoid, further strengthening the bridge between the algebraic behavior of monoids and rewriting theory.