Completely Regular Semigroups for Which Every Congruence on Each 𝒟-Class Extends to a Congruence on the Whole Semigroup

Abstract

In 1999, M. Petrich and N. R. Reilly proposed some problems in their monograph “Completely regular semigroups,” one of which is: Characterize those completely regular semigroups S with the property that every congruence on every 𝒟-class extends to a congruence on S. The problem is solved in this paper.

Key Words:

• Congruences
• Cryptogroups
• Left quasinormal cryptogroups
• Normal cryptogroups
• Regular cryptogroups

2010 Mathematics Subject Classification:

• 20M17

ACKNOWLEDGMENTS

We would like to express our profound gratitude to the referee for the valuable remarks and suggestions, which greatly improve the presentation of this article. Thanks also go to Dr. Mark Kambites for the timely communications.

Notes

Communicated by M. Kambites.