Combinatorial Results for Semigroups of Order-Preserving and A-Decreasing Finite Transformations

Abstract

For $n\in \mathbb {N}$, let $O_{n}$ be the semigroup of all order-preserving transformations on the finite chain $X_{n}=\{1,\ldots ,n\}$, under its natural order. For any non-empty subset A of $X_{n}$, let $O_{n}(A)$ and $O_{n}^+(A)$ be the subsemigroups of all order-preserving and A-decreasing, and of all order-preserving and A-increasing transformations on $X_{n}$, respectively. In this paper we obtain formulae for the number of elements and for the number of idempotents in $O_{n}(A)$. Moreover, we show that $O_{n}(A)$ contains a zero element if and only if $1\in A$, and then we obtain the number of nilpotents in $O_{n}(A)$ when $1\in A$.