On semigroups of transformations that preserve a double direction equivalence

For a non-empty set X, denote the full transformation semigroup on X by T(X) and suppose that E is an equivalence relation on X. Evidently, T-E*(X) = {alpha is an element of T(X)|(x y) is an element of E if and only if, (x alpha, y alpha) is an element of E for all, x y is an element of X, is a subsemigroup of T(X). In this article, we investigate Green relations, Green *-relations and Green (similar to)-relations, various kinds of regularities, F -abundant and G-abundant elements and left and right magnifying elements in T-E*(X). More specifically, we first obtain the necessary and sufficient conditions under which L (respectively, L *, (L) over tilde, R, R*, and (R) over tilde) is (left, right) compatible, R = R * or L = (L) over tilde. Then, we give the sufficient and necessary conditions such that T-E*(X) E is left regular (respectively, right regular, completely regular, intraregular, and completely simple). Finally, we characterize the F-abundant (respectively, G-abundant) and left (respectively, right) magnifying elements in T-E*(X)