The structure of -inverse semigroups

ABSTRACT

As a generalization of left inverse semigroups in the class of regular semigroups, we consider a class of semigroups which we name $\widetilde{\mathcal{ℒ}}$-inverse semigroups. After introducing the notion of left circle product for semigroups, we give a construction method of such a semigroups. It is proved that a semigroup S is an $\widetilde{\mathcal{ℒ}}$-inverse semigroup if and only if S can be expressed as a left circle product of an E-ample semigroup and a left regular band. Our work may be regarded as extending the result of Yamada for left inverse semigroups and the structure theorem obtained by Ren-Shum for ℒ*-inverse semigroups.

KEYWORDS:

• ℒ̃-inverse semigroups
• left circle products
• U-abundant semigroups
• weakly U-ample semigroups

2000 AMS MATHEMATICS SUBJECT CLASSIFICATION:

• 20M10

Acknowledgments

The authors would like to thank the referees for their very valuable modifying suggestions.