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Semigroup Forum

, 83:134 | Cite as

The regular part of a semigroup of transformations with restricted range

  • Jintana SanwongEmail author
RESEARCH ARTICLE

Abstract

Let T(X) be the full transformation semigroup on the set X and let T(X,Y) be the semigroup consisting of all total transformations from X into a fixed subset Y of X. It is known that
$$F(X, Y)=\{\alpha\in T(X, Y): X\alpha\subseteq Y\alpha\},$$
is the largest regular subsemigroup of T(X,Y) and determines Green’s relations on T(X,Y). In this paper, we show that F(X,Y)≅T(Z) if and only if X=Y and |Y|=|Z|; or |Y|=1=|Z|, and prove that every regular semigroup S can be embedded in F(S 1,S). Then we describe Green’s relations and ideals of F(X,Y) and apply these results to get all of its maximal regular subsemigroups when Y is a nonempty finite subset of X.

Keywords

Transformation semigroup Green’s relations Regular semigroup 

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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceChiang Mai UniversityChiangmaiThailand
  2. 2.Material Science Research Center, Faculty of ScienceChiang Mai UniversityCiangmaiThailand

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