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 \(1\in A\).