Axiomatizability and model completeness of the class of regular polygons (Q1897976)

Let \(S\) be a monoid with the unity element 1. An algebraic system \(\langle A; s \rangle_{s \in S}\) is a (left) polygon (or an \(S\)- polygon) provided that \(s_1 (s_2a) = (s_1s_2)a\) and \(1a = a\) for all \(s_1, s_2 \in S\) and \(a \in A\). Such a polygon \(\langle A;s \rangle_{s \in S}\) is denoted by \(_sA\). A polygon \(_sA\) is regular provided that, for every \(a \in A\), there exists an idempotent \(e \in S\) such that \(_sSa \cong {}_s Se\). The author describes monoids with axiomatizable and model complete classes of regular polygons.