On certain subsemigroups of finite oriented and order-decreasing full transformations

Abstract: Let $\mathcal{ORD}{n}$ be the semigroup consisting of all oriented and order-decreasing full transformations on the finite chain $X{n}={ 1<\cdots<n }$, and for $1\leq r\leq n-1$, let $$\mathcal{ORD}(n,r) ={\alpha \in \mathcal{ORD}_{n}\, :\, \lvert \textrm{im}(\alpha )\rvert \leq r}.$$ In this paper, we determine the cardinality of $\mathcal{ORD}(n,r)$ and the number of nilpotent elements of $\mathcal{ORD}(n,r)$, we find a minimal generating set and the rank of $\mathcal{ORD}(n,r)$, and moreover, we characterize all maximal subsemigroups of $\mathcal{ORD}(n,r)$ for each $3\leq r\leq n-1$.