ABSTRACT. This work is part of a research project aiming at developing rewriting methods to study diagrammatic algebras. In particular, we are interested in this work to diagrammatic algebras which can be seen as linear~(2,2) categories with an additional structure, for instance given by braidings, adjunctions or duals.

We present new termination heuristics for linear (3,2)-polygraphs presenting these linear (2,2)-categories based on the definition of an order similar to a monomial order, but that is not required to be total. This order is defined by counting the generators on the diagrams, and finding characteristics which are stable by contexts. We illustrate these methods by proving termination of a particular linear (3,2)-polygraph presenting a candidate 2-category for categorifying a quantum group.