### Factorizable Semigroups of Self-{\huge$\rho$}-Preserving Transformations

#### Abstract

Let $\mathcal{T}(X)$ be the semigroup of full transformations on a set $X$. For an equivalence $\rho$ on $X$, let $\mathcal{T}(X,\rho)=\{\alpha\in\mathcal{T}(X) : \forall a, b\in X, (a,b)\in\rho\Rightarrow(a\alpha,b\alpha)\in\rho\}$, and $\mathcal{T}(X,X/\rho)=\{\alpha\in\mathcal{T}(X) : \forall a\in X, (a,a\alpha)\in\rho\}$. Then $\mathcal{T}(X,\rho)$ is a subsemigroup of $\mathcal{T}(X)$ and $\mathcal{T}(X,X/\rho)$ is a subsemigroup of $\mathcal{T}(X,\rho)$. The set $\mathcal{T}(X,\rho)$ and $\mathcal{T}(X, X/\rho)$ are called the transformation semigroup preserving $\rho$, and the self-$\rho$-preserving transformation semigroup on $X$, respectively. For a semigroup $S$, $S$ is factorizable if $S = GE$ where $G$ is a subgroup of $S$ and $E$ is the set of all idempotents of $S$. Factorization of full transformation semigroups have been studied by Tirasupa in \cite{Tirasupa}. In this paper, we extend that work to $\mathcal{T}(X,\rho)$ and $\mathcal{T}(X,X/\rho)$ the transformation semigroups preserving equivalence relation $\rho$ and the self-$\rho$-preserving transformation semigroup on $X$.

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