On the nilpotents of the semigroup of partial contractions of a finite chain

Abstract

Let [n] = {1, 2, . . . , n} denote a finite chain. Consider the semigroups Pn and OIn, consisting of all partial, and all injective and order-preserving transformations on [n], respectively. Furthermore, let CPn = {α ∈ Pn : |xα − yα| ≤ |x − y| ∀ x, y ∈ Dom α} and OCIn = {α ∈ OIn : |xα − yα| ≤ |x − y| ∀ x, y ∈ Dom α}. It follows that CPn and OCIn are subsemigroups of Pn and OIn, respectively. In this paper, we characterize the nilpotent elements in CPn, compute the number of nilpotents of height (n − 1) in CPn, and determine the number of nilpotents in OCIn of a specific type