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：本文介绍了幺半群重写系统（MRS），这是字符串重写的一种抽象，其中约减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）。这一转变的部分动机源于逻辑：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 $\mathbf{NCRS_2}$ 为诺特且合流的 MRS 构成的 2-范畴。随后，我们证明了 $\mathbf{NCRS_2}$ 与 $\mathbf{Mon}$（幺半群范畴）之间存在一个典范的双伴随函子。最后，我们对所有能表示一个给定固定幺半群的诺特合流 MRS 进行了分类。为此，我们引入了广义初等蒂策变换（GETTs），并证明了幺半群的任意两个表示均可通过一个（可能无限的）此类变换序列相连接，从而得到了生成系统在 GETT 等价下的完整刻画。