In this note we present a sketch of a correspondence between hypergraph rewrite systems and higher-arity algebras. Our results have implications in a wide range of research areas, from diagrammatic calculus in category theory, to applied spectral methods in complex networks. Our key observation is that a choice of motif (model subhypergraph that determines a rewrite rule), induces a canonical algebraic operation on adjacency hypermatrices (higher-order tensors faithfully encoding hyperedge data). This is a direct generalization of the well-known fact that path-adjacency in graphs induces ordinary matrix multiplication. The rewrite perspective proves particularly illuminating when dealing with long-standing problems in algebraic hypergraph theory and higher-arity algebra such as higher-order sequentiality and generalized associativity.