Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {K}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">K</mi> </math></EquationSource> </InlineEquation> be one of the Green relations <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">R</mi> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {L}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">L</mi> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {D}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">D</mi> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {J}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">J</mi> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal {H}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation>. A <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathcal {K}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">K</mi> </math></EquationSource> </InlineEquation>-class is regular if all its elements are regular. The <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathcal {K}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">K</mi> </math></EquationSource> </InlineEquation>-classes of a semigroup are ruled by its regular ones if the triviality of all regular <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\mathcal {K}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">K</mi> </math></EquationSource> </InlineEquation>-classes implies the triviality of all these classes. This property is satisfied by finite semigroups, but also by compact semigroups and group-bound semigroups. The goal of this article is to provide a unified presentation of these last two results. To this end, we introduce a new class of semigroups, called safe semigroups, containing both compact semigroups and group-bound semigroups, for which <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathcal {D}= \mathcal {J}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">D</mi> <mo>=</mo> <mi mathvariant="script">J</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\mathcal {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">R</mi> </math></EquationSource> </InlineEquation>-classes, <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\mathcal {L}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">L</mi> </math></EquationSource> </InlineEquation>-classes and <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\mathcal {J}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">J</mi> </math></EquationSource> </InlineEquation>-classes are ruled by regular ones. The slightly more restricted class of secure semigroups makes it possible to also cover the case of <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\mathcal {H}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation>-classes.

Green classes ruled by regular ones

期刊论文
发表日期：

2026年5月19日

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Jean Éric Pin

摘要

Let $\mathcal {K}$ $K$ be one of the Green relations $\mathcal {R}$ $R$ , $\mathcal {L}$ $L$ , $\mathcal {D}$ $D$ , $\mathcal {J}$ $J$ or $\mathcal {H}$ $H$ . A $\mathcal {K}$ $K$ -class is regular if all its elements are regular. The $\mathcal {K}$ $K$ -classes of a semigroup are ruled by its regular ones if the triviality of all regular $\mathcal {K}$ $K$ -classes implies the triviality of all these classes. This property is satisfied by finite semigroups, but also by compact semigroups and group-bound semigroups. The goal of this article is to provide a unified presentation of these last two results. To this end, we introduce a new class of semigroups, called safe semigroups, containing both compact semigroups and group-bound semigroups, for which $\mathcal {D}= \mathcal {J}$ $D = J$ and $\mathcal {R}$ $R$ -classes, $\mathcal {L}$ $L$ -classes and $\mathcal {J}$ $J$ -classes are ruled by regular ones. The slightly more restricted class of secure semigroups makes it possible to also cover the case of $\mathcal {H}$ $H$ -classes.