Regularity and Green’s relations for the semigroups of partial and full contractions of a finite chain

Abstract

Let [n]={1,2,…,n} be a finite chain and let Pn be the semigroup of partial transformations on [n]. Let CPn={α∈Pn:(for allx,y∈Domα)|xα−yα|≤|x−y|} be the semigroup of partial contractions on [n]. Then CPn is a subsemigroup of Pn. In this paper, in the second and third sections we give a necessary and sufficient condition for an element in CPn to be regular and characterize all the Green’s relations in the semigroup CPn, respectively. In the subsequent fourth and fifth sections we obtain analogous results for the subsemigroups of order preserving and/or order reversing contractions, and that of full contractions, respectively. Finally, in the last section, we characterize all the maximal subgroups of these semigroups.