Lattices, Semigroups, and Universal Algebra pp 203-210 | Cite as

# The Kernel of an Idempotent Separating Congruence on a Regular Semigroup

## Abstract

Let S be a regular semigroup and E(S) its set of idempotents. Let θ be an idempotent separating congruence on S. Traditionally by the *kernel* ker θ of θ we understand the union of the idempotent θ-classes (see e.g. [4]). For an idempotent separating congruence θ the idempotent θ-classes are groups, namely the θ-classes containing idempotents. The *kernel normal system* of θ considered by Preston in [6] is the set of these idempotent θ-classes and contains more information than the above mentioned ker θ, which, after all, is just a subset of S. In the following we shall adopt still another approach to the concept of the kernel of an idempotent separating congruence on a regular semigroup. We shall give a survey of some of the results obtained in collaboration with K. S. S. Nambooripad.

## Keywords

Inverse Semigroup Regular Semigroup Group Morphism Identity Transformation Split Extension## Preview

Unable to display preview. Download preview PDF.

## References

- 1.Nambooripad, K.S.S.,
*Structure of Regular Semigroups. I*, Memoirs Amer. Math. Soc. 244, Providence, 1979.Google Scholar - 2.Pastijn, F.,
*The biorder on the partial groupoid of idempotents of a semigroup*, J. Algebra 65 (1980), 147–187.MathSciNetCrossRefzbMATHGoogle Scholar - 3.Pastijn, F. and M. Petrich,
*Regular Semigroups as Extensions*, Research Notes in Mathematics 136, Pitman, Boston, 1985.Google Scholar - 4.Pastijn, F. and M. Petrich,
*Congruences on regular semigroups*, Trans. Amer. Math. Soc, 295 (1986), 607–633.MathSciNetCrossRefzbMATHGoogle Scholar - 5.Petrich, M.,
*Inverse Semigroups*, Wiley, New York, 1984.zbMATHGoogle Scholar - 6.Preston, G. B.,
*Inverse semigroups*, J. London Math. Soc, 29 (1954), 396–403.MathSciNetCrossRefzbMATHGoogle Scholar