Algebraic critical pair lemma

International audience ; Convergent rewriting systems on algebraic structures give methods to prove coherence results and compute homological invariants of these structures. These methods are based on higher-dimensional extensions of the critical pair lemma that characterizes local confluence from confluence of critical pairs. The analysis of local confluence of rewriting systems on algebraic structures, such as groups or linear algebras, is complicated because of the underlying algebraic axioms, and local confluence properties require additional termination conditions. In this work, we define the structure of algebraic polygraph modulo that formalizes the interaction between the rules of the rewriting system and the inherent algebraic axioms, and we show a critical pair lemma algebraic polygraphs. We deduce from this result a critical pair lemma for rewriting systems on algebraic structures specified by rewriting systems convergent modulo AC. As an illustration, we explicit our constructions on linear rewriting systems.