Combinatorial results for semigroups of orientation-preserving and order-decreasing transformations

Abstract

Let \({{\mathscr {O}}{\mathscr {P}}{\mathscr {D}}}_{n}\) be the semigroup consisting of all orientation-preserving and order-decreasing full transformations on the finite chain \(X_{n}=\{1<\cdots <n\}\) , and let \(N({{\mathscr {O}}{\mathscr {P}}{\mathscr {D}}}_{n})\) be the subsemigroup consisting of all nilpotent elements of \({{\mathscr {O}}{\mathscr {P}}{\mathscr {D}}}_{n}\) . Moreover, for \(1\le r\le n-1\) , let $$\begin{aligned} {{\mathscr {O}}{\mathscr {P}}{\mathscr {D}}}(n,r) =\{\alpha \in {{\mathscr {O}}{\mathscr {P}}{\mathscr {D}}}_{n}\,:\, |\textrm{im}\,(\alpha )|\le r\}, \end{aligned}$$ and let \(N({{\mathscr {O}}{\mathscr {P}}{\mathscr {D}}}(n,r))\) be the subsemigroup consisting of all nilpotent elements of \({{\mathscr {O}}{\mathscr {P}}{\mathscr {D}}}(n,r)\) . In this paper, we compute the cardinalities of \({{\mathscr {O}}{\mathscr {P}}{\mathscr {D}}}_{n}\) , \(N({{\mathscr {O}}{\mathscr {P}}{\mathscr {D}}}_{n})\) , \({{\mathscr {O}}{\mathscr {P}}{\mathscr {D}}}(n,r)\) and \(N({{\mathscr {O}}{\mathscr {P}}{\mathscr {D}}}(n,r))\) , and find their ranks. Moreover, for each idempotent \(\xi \) in \({{\mathscr {O}}{\mathscr {P}}{\mathscr {D}}}_{n}\) , we show that \({{\mathscr {O}}{\mathscr {P}}{\mathscr {D}}}_{n}(\xi )=\{\ \alpha \in {{\mathscr {O}}{\mathscr {P}}{\mathscr {D}}}_{n} \,\ \alpha ^{m}=\xi \,\, \text{ for } \text{ some } \,\, m\in {\mathbb {N}}\ \}\) is the maximal nilpotent subsemigroup of \({{\mathscr {O}}{\mathscr {P}}{\mathscr {D}}}_{n}\) with zero \(\xi \) , and we find its cardinality and rank.