Combinatorial results for semigroups of order-preserving mappings
Peter M. Higgins · Mathematical Proceedings of the Cambridge Philosophical Society · 1993
Consider the finite setXn= {1,2, …,n} ordered in the standard way. LetTndenote the full transformation semigroup onXn, that is, the semigroup of all mappings α:Xn→Xnunder composition. We shall call α order-preserving ifi≤jimpliesiα ≤jα fori,j∈Xn, and α is decreasing ifiα ≤ifor alli∈Xn. This paper investigates combinatorial properties of the semigroup O of all order-preserving mappings onXn, and of its subsemigroup C which consists of all decreasing and order-preserving mappings.