J. Korean Math. Soc. 2011; 48(2): 289-300
Printed March 1, 2011
https://doi.org/10.4134/JKMS.2011.48.2.289
Copyright © The Korean Mathematical Society.
Preeyanuch Honyam and Jintana Sanwong
Chiang Mai University, Chiang Mai University
Let $T(X)$ denote the semigroup (under composition) of transformations from $X$ into itself. For a fixed nonempty subset $Y$ of $X$, let $S(X, Y) = \{\alpha\in T(X) : Y\alpha \subseteq Y\}.$ Then $S(X, Y)$ is a semigroup of total transformations of $X$ which leave a subset $Y$ of $X$ invariant. In this paper, we characterize when $S(X, Y)$ is isomorphic to $T(Z)$ for some set $Z$ and prove that every semigroup $A$ can be embedded in $S(A^{1}, A)$. Then we describe Green's relations for $S(X, Y)$ and apply these results to obtain its group $\mathcal{H}$-classes and ideals.
Keywords: transformation semigroups, Green's relations, ideals
MSC numbers: 20M12, 20M20
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