On certain semigroups of contraction mappings of a finite chain

Abstract

Let[n] ={1,2, . . . , n} be a finite chain and let Pn (resp.,Tn) be the semigroup of partial transformations on[n] (resp., full transformations on[n]). Let CPn={α∈ Pn: (for allx, y ∈ Dom α)|xα−yα|⩽|x−y|}(resp., CTn={α∈ Tn: (for allx, y∈[n])|xα−yα|⩽|x−y|}) be the subsemigroup of partial contractionmappings on[n](resp., subsemigroup of full contraction mappingson[n]). We characterize all the starred Green’s relations on C Pn and it subsemigroup of order preserving and/or order reversingand subsemigroup of order preserving partial contractions on[n], respectively. We show that the semigroups CPn and CTn, and some of their subsemigroups are left abundant semigroups for all n but not right abundant forn⩾4. We further show that the set ofregular elements of the semigroup CTn and its subsemigroup of order preserving or order reversing full contractions on[n], each formsa regular subsemigroup and an orthodox semigroup, respectively.