From Anticommutative Algebras to Combinatorial Structures: A Graph-Theoretic Approach

This article develops a combinatorial and graph-theoretic framework for the study of finite-dimensional anticommutative algebras. Given a fixed basis, we associate to each algebra a directed graph that may contain filled oriented triangles, encoding the pattern of nonzero products and the constraints imposed by anticommutativity. Focusing on three-dimensional algebras, we provide the complete list of combinatorial structures associated with each algebra in the known classification. Moreover, we develop two algorithmic procedures: the first determines whether a given anticommutative algebra satisfies the Tortkara identity, and the second identifies the additional conditions a given combinatorial structure must satisfy to correspond to a Tortkara algebra. Finally, we carry out a computational study of these algorithms. Our approach establishes a clear link between algebraic identities and discrete combinatorial objects, opening new avenues for the graphical analysis and classification of anticommutative algebras.