  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.
