Hyperidentities in associative graph algebras (Q2717941)

Graph algebras were introduced by \textit{C. R. Shallon} in her Ph.D. Thesis [Nonfinitely based binary algebras derived from lattices. UCLA, Los Angeles, Calif. (1979)] as a fruitful source of finite algebras whose laws do not have a finite basis, but they have proved useful in other contexts. Given a directed graph \(G\), with vertex set \(V\) and edge set \(E\), an algebra with a binary operation is defined on \(V\cup \{\infty\}\) by putting \(uv=u\) if \((u,v)\in E\) and letting all other products equal \(\infty\). Clearly, requiring that the algebra satisfies a certain identity places restrictions on the underlying graph. In this paper the algebras are required to satisfy the associative law, which places strong conditions on the graph, for example, the set of out-neighbours of any vertex has to induce a complete graph. The main theorem of this paper yields a classification of the hyperidentities of the class of associative graph algebras.