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arXiv · 2601.10564

Rewriting Systems on Arbitrary Monoids

Eduardo Magalhães
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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.

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Eduardo Magalhães. 2026-01-15. Rewriting Systems on Arbitrary Monoids. https://arxiv.org/abs/2601.10564

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