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.

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

“…Recently, Ali et al, (2018) extend this study to the general semigroup of partial contractions CPn. These semigroups were shown to be left abundant, for example see the work of Umar and Zubairu (Umar and Zubairu, 2018a;Umar and Zubairu, 2018b). In another development, Garba et al, (2017) characterized the starred Green's equivalences on the semigroup of full contraction CTn.…”