In traditional rewriting theory, one studies a set of terms up to a set of rewriting relations. In algebraic rewriting, one instead studies a vector space of terms, up to a vector space of relations. Strikingly, although both theories are very similar, most results (such as Newman's Lemma) require different proofs in these two settings. In this paper, we develop rewriting theory internally to a category $\mathcal C$ satisfying some mild properties. In this general setting, we define the notions of termination, local confluence and confluence using the notion of reduction strategy, and prove an analogue of Newman's Lemma. In the case of $\mathcal C= \operatorname{Set}$ or $\mathcal C = \operatorname{Vect}$ we recover classical results of abstract and algebraic rewriting in a slightly more general form, closer to von Oostrom's notion of decreasing diagrams.

中文翻译：

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抽象重写内化

在传统的重写理论中，人们研究一组术语直到一组重写关系。在代数重写中，人们改为研究项的向量空间，直至关系的向量空间。引人注目的是，尽管两种理论非常相似，但大多数结果（例如纽曼引理）在这两种设置中都需要不同的证明。在本文中，我们将内部重写理论发展为满足一些温和性质的类别 $\mathcal C$。在这种一般情况下，我们使用归约策略的概念定义了终止、局部汇合和汇合的概念，并证明了纽曼引理的类似物。在 $\mathcal C= \operatorname{Set}$ 或 $\mathcal C = \operatorname{Vect}$ 的情况下，我们以稍微更一般的形式恢复抽象和代数重写的经典结果，更接近 von Oostrom'