Green’s Relations and Natural Partial Order on the Regular Subsemigroup of Transformations Preserving an Equivalence Relation and Fixed a Cross-Section: Nares Sawatraksa, Chaiwat Namnak, Kritsada Sangkhanan - Team, Support (Thai Journal of Mathematics, 2019)
Let X be an arbitrary nonempty set and T(X) the full transformation semigroup on a set X. For an equivalence relation E on X and a cross-section R of the partition X/E induced by E, let \[ T_E(X, R) = {α T(X) : Rα = R and x,y X, (x, y) E (xα, yα) E }. \] Then the set Reg(T_E(X, R)) of all regular elements of T_E(X, R) is a regular subsemigroup of T(X). In this paper, we describe Green's relations for elements of the semigroup Reg(T_E(X, R)). Also, we discuss the natural partial order on semigroup of Reg(T_E(X, R)) and characterize when two elements of Reg(T_E(X, R)) are related under this order.