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The Kernel of an Idempotent Separating Congruence on a Regular Semigroup

  • Francis J. Pastijn

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 
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References

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

Copyright information

© Springer Science+Business Media New York 1990

Authors and Affiliations

  • Francis J. Pastijn
    • 1
  1. 1.Dept. of Mathematics, Statistics and Computer ScienceMarquette UniversityMilwaukeeUSA

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