Quasi-idempotents in finite semigroup of full order-preserving transformations

Abstract

Let X_n be the finite set {1,2,...,n} and O_n,r = {α∈O_n: |imα| ≤ r} (where 1 ≤ r ≤ n-1) be the ideals of a semigroup of full order-preserving transformations on the finite set. In this article, a study of quasi-idempotent elements via idempotents and generating set in the ideals of finite semigroup of full order-preserving transformations is carried out. The quasi-idempotent elements are characterised in this semigroup and that it is quasi-idempotent generated via idempotents. Moreover, the minimum length of a factorisations into products of quasi-idempotent elements is obtained.