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Abstract <p>We find the necessary and sufficient conditions for an act <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10502_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(X\)</EquationSource> <!--RusMath2570046Kozhukhov-m1--> </InlineEquation> over a trivial semigroup <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10502_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">\(\{ e\} \)</EquationSource> <!--RusMath2570046Kozhukhov-m2--> </InlineEquation> to be Cantor (or co-Cantor), that is, for any act <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10502_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(Y\)</EquationSource> <!--RusMath2570046Kozhukhov-m3--> </InlineEquation> the existence of injective (respectively, surjective) homomorphisms <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10502_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">\(X \to Y\)</EquationSource> <!--RusMath2570046Kozhukhov-m4--> </InlineEquation> and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10502_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">\(Y \to X\)</EquationSource> <!--RusMath2570046Kozhukhov-m5--> </InlineEquation> implies the isomorphism <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10502_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">\(X \cong Y\)</EquationSource> <!--RusMath2570046Kozhukhov-m6--> </InlineEquation>.</p>

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Conditions of Cantor and Co-Cantor Properties for Acts over the Trivial Semigroup

  • 期刊论文

  • 发表日期：

    2025年6月1日

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  • I. B. Kozhukhov,
  • A. S. Sotov

摘要

Abstract

We find the necessary and sufficient conditions for an act \(X\) over a trivial semigroup \(\{ e\} \) to be Cantor (or co-Cantor), that is, for any act \(Y\) the existence of injective (respectively, surjective) homomorphisms \(X \to Y\) and \(Y \to X\) implies the isomorphism \(X \cong Y\) .