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<p>In 1966, Mal’cev proved that a class <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12_2025_902_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal {K}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">K</mi> </math></EquationSource> </InlineEquation> of first-order structures with a specified signature is a quasivariety if and only if <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12_2025_902_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal {K}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">K</mi> </math></EquationSource> </InlineEquation> contains a unit and is closed under isomorphic images, substructures, and reduced products. In this article, we present a proof of this theorem in <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12_2025_902_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(\textsf{ZF}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="sans-serif">ZF</mi> </math></EquationSource> </InlineEquation> (i.e., the Zermelo–Fraenkel set theory without the axiom of choice).</p>

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

  • 期刊论文

  • 发表日期：

    2025年8月21日

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  • Guozhen Shen

摘要

In 1966, Mal’cev proved that a class \(\mathcal {K}\) K of first-order structures with a specified signature is a quasivariety if and only if \(\mathcal {K}\) 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 \(\textsf{ZF}\) ZF (i.e., the Zermelo–Fraenkel set theory without the axiom of choice).