# SEMIGROUPS OF TRANSFORMATIONS WITH INVARIANT SET

• Published : 2011.03.01

#### Abstract

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 H-classes and ideals.

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#### Cited by

1. On certain order-preserving transformation semigroups vol.45, pp.7, 2017, https://doi.org/10.1080/00927872.2016.1233333
2. Semigroups of transformations with fixed sets vol.36, pp.1, 2013, https://doi.org/10.2989/16073606.2013.779958
3. REGULARITY OF TRANSFORMATION SEMIGROUPS DEFINED BY A PARTITION vol.31, pp.2, 2016, https://doi.org/10.4134/CKMS.2016.31.2.217
4. NATURAL PARTIAL ORDER IN SEMIGROUPS OF TRANSFORMATIONS WITH INVARIANT SET vol.87, pp.01, 2013, https://doi.org/10.1017/S0004972712000287
5. Magnifying elements of semigroups of transformations with invariant set pp.1793-7183, 2018, https://doi.org/10.1142/S1793557119500566