Adequate Transversals of Quasi-Adequate Semigroups

Abstract

The concept of an adequate transversal of an abundant semigroup was introduced by El-Qallali in [8] whilst in [7], he and Fountain initiated the study of quasi-adequate semigroups as natural generalisations of orthodox semigroups. In this work we provide a structure theorem for adequate transversals of certain types of quasi-adequate semigroup and from this deduce Saito's classic result on the structure of inverse transversals of orthodox semigroups. We also apply it to left ample adequate transversals of left adequate semigroups and provide a structure for these based on semidirect products of adequate semigroups by left regular bands.

Key Words:

• Abundant semigroup
• Adequate
• Adequate transversal
• Inverse transversal
• Left adequate
• Left ample
• Semidirect product
• Quasi-adequate
• Regular semigroup

2000 Mathematics Subject Classification:

• 20M10

ACKNOWLEDGMENT

The authors would like to thank V. Gould and J. Fountain for useful discussions relating to this work.

Notes

Communicated by V. Gould.