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Semigroup Forum

, Volume 82, Issue 2, pp 307–318 | Cite as

Regular centralizers of idempotent transformations

  • Jorge André
  • João AraújoEmail author
  • Janusz Konieczny
Research Article

Abstract

Denote by T(X) the semigroup of full transformations on a set X. For εT(X), the centralizer of ε is a subsemigroup of T(X) defined by C(ε)={αT(X):αε=εα}. It is well known that C(id X )=T(X) is a regular semigroup. By a theorem proved by J.M. Howie in 1966, we know that if X is finite, then the subsemigroup generated by the idempotents of C(id X ) contains all non-invertible transformations in C(id X ).

This paper generalizes this result to C(ε), an arbitrary regular centralizer of an idempotent transformation εT(X), by describing the subsemigroup generated by the idempotents of C(ε). As a corollary we obtain that the subsemigroup generated by the idempotents of a regular C(ε) contains all non-invertible transformations in C(ε) if and only if ε is the identity or a constant transformation.

Keywords

Idempotent transformations Regular centralizers Generators 

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Jorge André
    • 1
  • João Araújo
    • 1
    • 2
    Email author
  • Janusz Konieczny
    • 3
  1. 1.Centro de ÁlgebraUniversidade de LisboaLisboaPortugal
  2. 2.Universidade AbertaLisboaPortugal
  3. 3.Department of MathematicsUniversity of Mary WashingtonFredericksburgUSA

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