String diagrams constitute an intuitive and expressive graphical syntax that has found application in a very diverse range of fields including concurrency theory, quantum computing, control theory, machine learning, linguistics, and digital circuits. Rewriting theory for string diagrams relies on a combinatorial interpretation as double-pushout rewriting of certain hypergraphs. As previously studied, there is a `tension' in this interpretation: in order to make it sound and complete, we either need to add structure on string diagrams (in particular, Frobenius algebra structure) or pose restrictions on double-pushout rewriting (resulting in `convex' rewriting). From the string diagram viewpoint, imposing a full Frobenius structure may not always be natural or desirable in applications, which motivates our study of a weaker requirement: commutative monoid structure. In this work we characterise string diagram rewriting modulo commutative monoid equations, via a sound and complete interpretation in a suitable notion of double-pushout rewriting of hypergraphs.

翻译：字符串图构成了一种直观且富有表现力的图形语法，已被广泛应用于并发理论、量子计算、控制理论、机器学习、语言学和数字电路等多个领域。字符串图的重写理论依赖于将其解释为某些超图的双推重写。正如先前研究所揭示，这种解释中存在一种"张力"：为了使其可靠且完备，我们要么需要在字符串图上添加结构（特别是Frobenius代数结构），要么对双推重写施加限制（产生"凸"重写）。从字符串图的角度看，在应用中施加完全Frobenius结构并不总是自然或理想的，这促使我们研究一种更弱的要求：交换幺半群结构。本文通过将字符串图模交换幺半群方程的重写，可靠且完备地解释为一种适当的超图双推重写，从而对其进行刻画。