Axiomatizability by ${{\forall}{\exists}!}$-sentences

Abstract

A ${\forall\exists!}$-sentence is a sentence of the form ${\forall x_{1}\cdots x_{n}\exists!y_{1}\cdots y_{m}O(\overline{x},\overline{y})}$, where O is a quantifier-free formula, and ${\exists!}$ stands for “there exist unique”. We prove that if ${\mathcal{C}}$ is (up to isomorphism) a finite class of finite models then ${\mathcal{C}}$ is axiomatizable by a set of ${\forall\exists!}$-sentences if and only if ${\mathcal{C}}$ is closed under isomorphic images, ${\mathcal{C}}$ has the intersection property, and ${\mathcal{C}}$ is closed under fixed-point submodels. This result is employed to characterize the subclasses of finitely generated discriminator varieties axiomatizable by sentences of the form ${\forall\exists!\bigwedge p=q}$.