ON THE STRUCTURE OF REGULAR CRYPTO SEMIGROUPS

Abstract

Superabundant semigroups are generalizations of completely regular semigroups written the class of abundant semigroups. It has been shown by Fountain that an abundant semigroup is superabundant if and only if it is a semilattice of completely J ^*-simple semigroups. Reilly and Petrich called a semigroup S cryptic if the Green's relation H is a congruence on S. In this paper, we call a superabundant semigroup S a regular crypto semigroup if H ^* is a congruence on S such that S/H ^* is a regular band. It will be proved that a superabundant semigroup S is a regular crypto semigroup if and only if S is a refined semilattice of completely J ^*-simple semigroups. Thus, regular crypto semigroups are generalization of the cryptic semigroups as well as abundant semigroups.

Keywords:

• Crypto semigroups
• Regular bands
• Refined semilattice
• 1991 Mathematics Subject Classification: 20M10, 06B10

ACKNOWLEDGMENT

The authors would like to thank Professors Guo Yuqi and Zhang Liang for their helpful comments given to this paper. The first author is particularly grateful to his advisor Professor Guo Yuqi for his encouragement and guidance in his research work. The research is partially supported by a UGC(HK) earmarked grant #2260126(1999/2001).