Abundant subsemigroups of order-preserving transformations semigroup

Let Pn , T and On be the partial transformations, the (full) transformations and order-preserving (singular) transformations semigroups on the set X n n = {1,2,..., } , respectively. Moreover, for any non-empty subset of Xn , let On n (A) = {a a ÎO : ("x Î £ A) x x} be the subsemigroup of all order-preserving and A-decreasing (singular) transformations on the set Xn . In this study, first we study locally maximal idempotent-generated subsemigroups of order-preserving transformations semigroup. Moreover, we examine the * - Green's equivalences of ( ) O A n and show ( ) O A n is an abundant subsemigroup of order-preserving transformations semigroup. Finally, we study generating sets and ranks of ( ) O A n and show its idempotent rank is 2n - 2 - \ A n , its rank is n -1 if 1ÎA and n otherwise. Key Words: Order-preserving transformation, A-decreasing transformation, abundant subsemigroup, generating set, rank