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arXiv · 2501.00766

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

Guozhen Shen
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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).

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Guozhen Shen. 2025-01-01. A choice-free proof of Mal'cev's theorem on quasivarieties. https://doi.org/10.1007/s00012-025-00902-x

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