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The regular part of a semigroup of transformations with restricted range

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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.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Chiang Mai University, Chiangmai, 50200, Thailand

    Jintana Sanwong

  2. Material Science Research Center, Faculty of Science, Chiang Mai University, Ciangmai, Thailand

    Jintana Sanwong

Authors
  1. Jintana Sanwong
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Correspondence to Jintana Sanwong.

Additional information

Communicated by Thomas E. Hall.

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Sanwong, J. The regular part of a semigroup of transformations with restricted range. Semigroup Forum 83, 134–146 (2011). https://doi.org/10.1007/s00233-011-9320-z

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  • DOI: https://doi.org/10.1007/s00233-011-9320-z

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