The class of embeddable semigroup amalgams is not finitely axiomatizable (Q1380049)

Let \(\{S_i\mid i\in I\}\) be a family of semigroups, \(U\) a subsemigroup of \(S_i\) for all \(i\in I\) and \(S_i\cap S_j=U\) for all \(i,j\in I\), \(i\neq j\). The semigroup amalgam \({\mathcal A}=[\{S_i\mid i\in I\};U]\) determines a partial groupoid \({\mathcal G}({\mathcal A})\) on \({\mathcal G}=\bigcup\{S_i\mid i\in I\}\) in which the product of two elements is defined if and only if they both lie in the same \(S_i\) and their product in \(\mathcal G\) coincides with their product in \(S_i\). A semigroup amalgam \(\mathcal A\) is called embeddable in a semigroup if the partial groupoid \({\mathcal G}({\mathcal A})\) is embeddable in a semigroup. The author proves that the class of all embeddable semigroup amalgams is not first-order finitely axiomatizable.