A choice-free proof of Mal’cev’s theorem on quasivarieties

Abstract

In 1966, Mal’cev proved that a class \(\mathcal {K}\) of first-order structures with a specified signature is a quasivariety if and only if \(\mathcal {K}\) contains a unit and is closed under isomorphic images, substructures, and reduced products. In this article, we present a proof of this theorem in \(\textsf{ZF}\) (i.e., the Zermelo–Fraenkel set theory without the axiom of choice).