On nonstandard axiomatization of elementarily nonaxiomatizable classes of discrete algebraic systems (Q1972198)

The article under review continues the author's study of the properties of classes of algebraic systems defined by nonstandard first-order formulas (see the original paper for references and definitions). The author proves that a class of algebraic systems of a finite signature is axiomatizable by means of nonstandard identities if and only if it is closed under subsystems, homomorphic images, and finite cartesian products, which is one of the nonstandard versions of Birkhoff's Theorem. A similar result is proven for nonstandard quasivarieties: A class of algebraic systems of a finite signature is axiomatizable by means of nonstandard quasi-identities if and only if it is closed under subsystems and finite cartesian products.