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<p>We determine the structure of the power graph of a completely regular semigroup and describe the automorphism group of power graph. Corresponding to Cameron and Ghosh’s problem about power graphs of groups, we prove that a completely simple semigroup whose automorphism group is the same as that of its power graph either is a left (or right) zero semigroup or is isomorphic to the Klein group of order 4. Further, we characterize the structure of the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathscr {J}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">J</mi> </math></EquationSource> </InlineEquation>-classes of a completely regular semigroup which satisfies the above condition and examine the interrelationships of these <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathscr {J}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">J</mi> </math></EquationSource> </InlineEquation>-classes. Finally, we completely characterize the structure of a normal cryptogroup satisfying this condition.</p>

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Power graphs of completely regular semigroups

  • 期刊论文

  • 发表日期：

    2025年9月17日

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  • Yanliang Cheng,
  • Yong Shao,
  • Lingli Zeng

摘要

We determine the structure of the power graph of a completely regular semigroup and describe the automorphism group of power graph. Corresponding to Cameron and Ghosh’s problem about power graphs of groups, we prove that a completely simple semigroup whose automorphism group is the same as that of its power graph either is a left (or right) zero semigroup or is isomorphic to the Klein group of order 4. Further, we characterize the structure of the \(\mathscr {J}\) J -classes of a completely regular semigroup which satisfies the above condition and examine the interrelationships of these \(\mathscr {J}\) J -classes. Finally, we completely characterize the structure of a normal cryptogroup satisfying this condition.