Quasi-equational theories of graphs of semigroups

In this lecture I would like to study the finite axiomatizability problem for semigroups from a bit different perspective. The graph of a semigroup S is the relational structure $G(S)=(S,R)$, where ternary relation $R$ is the graph of the multiplication in $S$. For a class $C$ of semigroups by $G(C)$ we denote the class of all graphs of members of $C$. By a quasi-identity we mean an universal closure of a formula $A1 \wedge \ldots \wedge A_n \longrightarrow A$, where $A_n, A$ are atomic formulas with predicates in $\{R,=\}$. A quasi-equational theory of $G(C)$ is the set of all quasi-identities valid in $G(C)$. Here the result follows: Let $C$ be a class of semigroups possessing a nontrivial member with a neutral element. Then the quasi-equational theory of $G(C)$ is not finitely axiomatizable.