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<p>The purpose of this note is to document that the study of <i>E</i>-solid completely 0-simple semigroups cannot be reduced to the study of maximal completely simple subsemigroups of such semigroups. An example of a finite <i>E</i>-solid completely 0-simple semigroup <i>S</i> is provided such that <i>S</i> does not belong to the variety of semigroups generated by the collection of all maximal completely simple subsemigroups of <i>S</i> together with the five-element combinatorial Brandt semigroup <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="233_2025_10544_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\(B_2\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>B</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>. This is in sharp contrast to the situation which is known to hold for arbitrary Brandt semigroups relatively to their maximal subgroups.</p>

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A counter-example in the theory of E-solid completely 0-simple semigroups

  • 期刊论文

  • 发表日期：

    2025年5月30日

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  • Jiří Kad’ourek

摘要

The purpose of this note is to document that the study of E-solid completely 0-simple semigroups cannot be reduced to the study of maximal completely simple subsemigroups of such semigroups. An example of a finite E-solid completely 0-simple semigroup S is provided such that S does not belong to the variety of semigroups generated by the collection of all maximal completely simple subsemigroups of S together with the five-element combinatorial Brandt semigroup \(B_2\) B 2 . This is in sharp contrast to the situation which is known to hold for arbitrary Brandt semigroups relatively to their maximal subgroups.