On certain semigroups of contraction mappings of a finite chain

Let [n] = {1, 2, ..., n} be a finite chain and let P-n (resp., T-n) be the semigroup of partial transformations on [n] (resp., full transformations on [n]). Let CPn = {alpha is an element of P-n : (for all x, y is an element of Dom alpha) vertical bar x alpha - y alpha vertical bar <= vertical bar x - y vertical bar} (resp., CTn = {alpha is an element of T-n : (for all x, y is an element of [n]) vertical bar x alpha - y alpha vertical bar <= vertical bar x - y vertical bar}) 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 CPn 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 CPn and CTn and some of their subsernigroups are left abundant semigroups for all n but not right abundant for n >= 4. We further show that the set of regular elements of the semigroup CTn and its subsemigroup of order preserving or order reversing full contractions on [n], each forms a regular subsemigroup and an orthodox semigroup, respectively.