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

TL;DR

This paper provides a choice-free proof of Mal'cev's theorem characterizing quasivarieties in first-order structures, avoiding reliance on the axiom of choice.

Contribution

It offers a novel proof of Mal'cev's theorem within ZF set theory, expanding understanding of quasivarieties without the axiom of choice.

Findings

Proof of Mal'cev's theorem in ZF set theory

Characterization of quasivarieties without choice

Enhanced foundational understanding of algebraic structures

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 isomorphisms, substructures, and reduced products. In this article, we present a proof of this theorem in $\mathsf{ZF}$ (the Zermelo--Fraenkel set theory without the axiom of choice).