Abundance and magnifying elements in transformation semigroups that preserve order and equivalence

Let rho be a nontrivial equivalence relation on a totally ordered set X and T(X,rho) be the semigroup of all transformations of X that preserve rho. In this paper, we consider a subsemigroup of T(X,rho) defined byO(X,rho) = {alpha is an element of T(X,rho):for all x , y is an element of X, x <= y double right arrow x alpha <= y alpha }and present a necessary and sufficient condition under which the semigroup O(X,rho) is abundant. Also, we describe all the left and right magnifying elements in O(X,rho).