Regularity and Green ’ s Relations for Generalized Semigroups of Transformations with Invariant Set

Let TX be the full transformation semigroup on a set X. For Y ⊆ X, the semigroup S (X, Y) = { f ∈ TX : f (Y) ⊆ Y} is a subsemigroup of TX . Fix an element θ ∈ S (X,Y) and for f , g ∈ S (X,Y), define a new operation ∗ on S (X,Y) by f ∗g = f θg where f θg denotes the produce of g, θ and f in the original sense. Under this operation, the semigroup S (X,Y) forms a semigroup which is called generalized semigroup of S (X,Y) with the sandwich function θ and denoted by S (X,Y, ∗θ). In this paper we first characterize the regular elements and then describe Green’s relations for the semigroup S (X,Y, ∗θ).

Let S be a semigroup and a, b ∈ S .If a = axa for some x ∈ S , then a is called a regular element of S .The semigroup S is called regular if all its elements are regular.If a and b generate the same left principle ideal, that is, S 1 a = S 1 b, then we say that a and b are L equivalent and write (a, b) ∈ L or a L b.If a and b generate the same right principle ideal, that is, aS 1 = bS 1 , then we say that a and b are R equivalent and write (a, b) ∈ R or a R b.If a and b generate the same principle ideal, that is, S 1 aS 1 = S 1 bS 1 , then we say that a and b are J equivalent and write (a, b) ∈ J or a J b.It is not difficult to see that L, R and J are equivalence relations on S .Let H = L ∩ R and D = L ∨ R. Then H and D are also equivalences.These five equivalences are usually called Green's relations on S .They were introduced by J.A. Green and play an important role in the study of the algebraic structure of semigroups.
Let T X be the full transformation semigroup on a set X. Given a subset Y of X, the authors in (Honyam, P. & Sanwong, J., 2011) observed a class of subsemigroup of T X defined by It is clear that if Y = X then S (X, Y) = T X .To this extent the semigroup S (X, Y) is regarded as a generalization of T X .Regularity for the elements in S (X, Y) and Green's relations on S (X, Y) were described in (Honyam, P. & Sanwong, J., 2011).
We apply transformations on the left so that for f, g ∈ S (X, Y), their product f g is the transformation obtained by first performing g and then f .Fix an element θ ∈ S (X, Y) and for f, g ∈ S (X, Y), define a new operation * on S (X, Y) by f * g = f θg where f θg denotes the produce of g, θ and f in the original sense.Under this operation, the semigroup S (X, Y) forms a semigroup which is called generalized semigroup of S (X, Y) with the sandwich function θ and denoted by S (X, Y, * θ ).Then S (X, Y, * θ ) = S (X, Y) as sets.Moreover, if θ = id X (the identity transformation on the set X), then S (X, Y, * θ ) = S (X, Y) as semigroups.The generalized transformation semigroups of the various subsemigroups of T X were studied by many authors, see for example (Hickey, J. B., 1983;Kemprasit, Y. & Jaidee, S., 2005;Magill, K. D. Jr. & Subbiah, S., 1975;Pei, H. S., Sun, L. & Zhai, H. C., 2007;Symons, J. S., 1975;Tsyaputa, G. Y., 2004).
The purpose of this paper is to investigate the regularity of elements and Green's relations on generalized semigroup S (X, Y, * θ ).Accordingly, in Section 2, the condition under which an element f ∈ S (X, Y, * θ ) is regular is analyzed.In Section 3, Green's relations on S (X, Y, * θ ) are considered and the relations L, R, H, D and J are descried for arbitrary elements, respectively.

The Regular Elements of S (X, Y, * θ )
In this section we investigate the condition under which an element of S (X, Y, * θ ) is regular.
Theorem 2.1.Let f ∈ S (X, Y, * θ ).Then f is regular if and only if the following statements hold.
(1) θ| f (X) is injective. ( Conversely, assume that (1)-( 2 Proof.Suppose that Reg(S (X, Y, * θ )) = Reg(S (X, Y)).Since the identity transformation id X on X is regular in S (X, Y), we have that id X is also regular in Conversely, we need to show that Reg(S (X, Y)) ⊆ Reg(S (X, Y, * θ )).For this purpose, let Theorem 2.3.The semigroup S (X, Y, * θ ) is regular if and only if the following statements hold.
Denote by π( f ) the partition of X induced by f ∈ T X , namely, ψ is said to be θ Y -admissible.If ψ is bijective and both ψ and ψ −1 are θ Y -admissible, then ψ is said to be θ * Y -admissible.Now we begin with the relation L in S (X, Y, * θ ).