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 TE^*(X) of $T(X)$ by\[TE^*(X)=\{α∈ T(X):for all\ x,y∈ X, (x,y)∈ E⇔ (xα,yα)∈ E\}.\]In this paper, we show that the regular part of TE^*(X), denoted (T), is the largest regular subsemigroup of TE^*(X). Then its Green's relations and ideals are described. Moreover, we find the kernel of (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.