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

Let ${\mathcal{O}\mathcal{R}\mathcal{D}}_n$ be the semigroup consisting of all oriented and order-decreasing full transformations on the finite chain $X_n=\{1<\cdots <n\}$, and for $1\le r\le n-1$, let $$\mathcal{O}\mathcal{R}\mathcal{D}(n,r)=\{\alpha \in {\mathcal{O}\mathcal{R}\mathcal{D}}_n:\mid \text{im}(\alpha )\mid \le r\}\mathrm{.}$$ In this paper, we determine the cardinality of $\mathcal{O}\mathcal{R}\mathcal{D}(n,r)$ and the number of nilpotent elements of $\mathcal{O}\mathcal{R}\mathcal{D}(n,r)$, we find a minimal generating set and the rank of $\mathcal{O}\mathcal{R}\mathcal{D}(n,r)$, and moreover, we characterize all maximal subsemigroups of $\mathcal{O}\mathcal{R}\mathcal{D}(n,r)$ for each $3\le r\le n-1$.