The regular part of transformation semigroups that preserve double direction equivalence relation

Let $T(X)$ be the full transformation semigroup on a set $X$ under the composition of functions. For any equivalence relation $E$ on $X$, define a subsemigroup $T_{E^{\ast }}(X)$ of $T(X)$ by $$T_{E^{\ast }}(X)=\{\alpha \in T(X):\text{for all }x,y\in X,(x,y)\in E\iff (x\alpha ,y\alpha )\in E\}\mathrm{.}$$ In this paper, we show that the regular part of $T_{E^{\ast }}(X)$, denoted $\mathrm{R}\mathrm{e}\mathrm{g}(T)$, is the largest regular subsemigroup of $T_{E^{\ast }}(X)$. Then its Green’s relations and ideals are described. Moreover, we find the kernel of $\mathrm{R}\mathrm{e}\mathrm{g}(T)$ which is a right group and can be written as a union of symmetric groups. Finally, we prove that every right group can be embedded in that kernel.