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<p>In this paper, we establish that for any homomorphism <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\varphi : S \rightarrow T\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>φ</mi> <mo>:</mo> <mi>S</mi> <mo stretchy="false">→</mo> <mi>T</mi> </mrow> </math></EquationSource> </InlineEquation> between two topological semigroups <i>S</i> and <i>T</i>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(S/\ker \varphi \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mo stretchy="false">/</mo> <mo>ker</mo> <mi>φ</mi> </mrow> </math></EquationSource> </InlineEquation> forms a topological semigroup if and only if <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\varphi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation> is <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\varphi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation>-saturated continuous. We provide necessary and sufficient conditions for the collection of all topological semigroup congruences on a topological semigroup to constitute a lattice. Furthermore, we outline the conditions under which a group congruence on a semigroup is a simple group congruence. Lastly, we define essential criteria for a group congruence on a topological semigroup to be classified as a topological group congruence.</p>

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Congruences on topological semigroups

  • 期刊论文

  • 发表日期：

    2025年8月29日

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  • Sunil Kumar Maity,
  • Monika Paul

摘要

In this paper, we establish that for any homomorphism \(\varphi : S \rightarrow T\) φ : S → T between two topological semigroups S and T, \(S/\ker \varphi \) S / ker φ forms a topological semigroup if and only if \(\varphi \) φ is \(\varphi \) φ -saturated continuous. We provide necessary and sufficient conditions for the collection of all topological semigroup congruences on a topological semigroup to constitute a lattice. Furthermore, we outline the conditions under which a group congruence on a semigroup is a simple group congruence. Lastly, we define essential criteria for a group congruence on a topological semigroup to be classified as a topological group congruence.