On certain semigroups of contraction mappings of a finite chain

Abstract

Let \([n]=\{1,2,\ldots,n\}\) be a finite chain and let \(\mathcal{P}_{n}\) (resp., \(\mathcal{T}_{n}\)) be the semigroup of partial transformations on \([n]\) (resp., full transformations on \([n]\)). Let \(\mathcal{CP}_{n}=\{\alpha\in \mathcal{P}_{n}: (\text{for all }x,y\in \operatorname{Dom}\alpha)\ |x\alpha-y\alpha|\leq|x-y|\}\) (resp., \(\mathcal{CT}_{n}=\{\alpha\in \mathcal{T}_{n}: (\text{for all }x,y\in [n])\ |x\alpha-y\alpha|\leq|x-y|\}\) ) be the subsemigroup of partial contraction mappings on \([n]\) (resp., subsemigroup of full contraction mappings on \([n]\)). We characterize all the starred Green's relations on \(\mathcal{CP}_{n}\) and it subsemigroup of order preserving and/or order reversing and subsemigroup of order preserving partial contractions on \([n]\), respectively. We show that the semigroups \(\mathcal{CP}_{n}\) and \(\mathcal{CT}_{n}\), and some of their subsemigroups are left abundant semigroups for all \(n\) but not right abundant for \(n\geq 4\). We further show that the set of regular elements of the semigroup \(\mathcal{CT}_{n}\) and its subsemigroup of order preserving or order reversing full contractions on \([n]\), each forms a regular subsemigroup and an orthodox semigroup, respectively.