On Ordered Semigroups Which are Semilattices of Simple and Regular Semigroups

Abstract

We characterize the ordered semigroups which are decomposable into simple and regular components. We prove that each ordered semigroup which is both regular and intra-regular is decomposable into simple and regular semigroups, and the converse statement also holds. We also prove that an ordered semigroup S is both regular and intra-regular if and only if every bi-ideal of S is an intra-regular (resp. semisimple) subsemigroup of S. An ordered semigroup S is both regular and intra-regular if and only if the left (resp. right) ideals of S are right (resp. left) quasi-regular subsemigroups of S. We characterize the chains of simple and regular semigroups, and we prove that S is a complete semilattice of simple and regular semigroups if and only if S is a semilattice of simple and regular semigroups. While a semigroup which is both π-regular and intra-regular is a semilattice of simple and regular semigroups, this does not hold in ordered semigroups, in general.

Key Words:

• Bi-ideal
• Chain of simple and regular semigroups
• Intra-regular
• Left (right) ideal
• Ordered semigroup
• Regular
• Right (left) quasi-regular
• Semisimple
• Semilattice (complete semilattice) of simple and regular semigroups

2010 Mathematics Subject Classification:

• 06F05
• 20M10
• 20M17

ACKNOWLEDGMENTS

We would like to express our best thanks to Professor Irena Swanson for editing, communicating the paper, and her prompt reply. Our best thanks to the referee also who read the paper very carefully, actually worked seriously on it.

Notes

Communicated by I. Swanson.