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.

翻译：本文引入幺半群重写系统（MRS），这是字符串重写的一种抽象，其中约简定义在任意环境幺半群上，而非自由幺半群（即词的自由幺半群）。这一转变部分源于逻辑动机：自由幺半群类不是一阶可公理化的，因此在将一阶方法应用于重写表示时，“在自由设定下工作”无法在系统内部处理。为了从范畴论角度分析这些系统，我们定义 $\mathbf{NCRS_2}$ 为诺特合流 MRS 的 2-范畴。随后证明 $\mathbf{NCRS_2}$ 与 $\mathbf{Mon}$ 之间存在典范双伴随关系。最后，我们对所有表示给定固定幺半群的诺特合流 MRS 进行分类。为此，我们引入广义初等蒂策变换（GETT），并证明幺半群的任意两个表示均可通过（可能无限的）一系列此类变换相互连接，从而在 GETT 等价意义下完全刻画了生成系统的分类。