On certain semigroups of contraction mappings of a finite chain

Let [n] = {1, 2, …, n} be a őnite chain and let P_n (resp., T_n ) be the semigroup of partial transformations on [n] (resp., full transformations on [n]). Let CP_n = {α ∈ P_n: (for all x, y ∈Domα) |xα − yα| ⩽ |x − y|} (resp., CT_n = {α ∈ T_n: (for all x, y ∈ [n]) |xα−yα| ⩽ |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 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 CP_n and CT_n, and some of their subsemigroups 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 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.