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

Let [n]={1,2,…,n} be a finite chain and let P_n be the semigroup of partial transformations on [n]. Let CP_n={α∈P_n:(for allx,y∈Domα)|xα−yα|≤|x−y|} be the semigroup of partial contractions on [n]. Then CP_n is a subsemigroup of P_n. In this paper, in the second and third sections we give a necessary and sufficient condition for an element in CP_n to be regular and characterize all the Green's relations in the semigroup CP_n, 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.