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 all x, y ∈ Dom α) |xα−yα| ≤ |x−y|} (resp. , CT n = {α ∈ Tn : (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 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 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.