On Rank of Semigroup of Order-Preserving Order-Decreasing Partial Contraction Mappings on a Finite Chain

Abstract

Let [n] = {1, 2, . . . , n} be a finite chain, and ODCPn be the semigroup of order-preserving and order-decreasing partial contraction mappings on [n]. In this paper, we study the rank properties of the two-sided ideals of ODCPn. We show that the rank of Kp = {α ∈ ODCPn : |im α| ≤ p}, for 2 ≤ p ≤ n, is and hence, the rank of ODCPn is 2n