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Keywords:
order-preserving transformation; orientation-preserving; (completely) isolated subsemigroup; generating set; rank
order-preserving transformation; orientation-preserving; (completely) isolated subsemigroup; generating set; rank
Summary:
Let $\mathcal {O}_{n}$ be the semigroup of all order-preserving (full) transformations on the finite chain $X_{n}=\{1,\ldots ,n\}$ under its natural order. For a singular idempotent $\xi $, it is shown that $\mathcal {O}_{n}(\xi )=\{ \alpha \in \mathcal {O}_{n} \colon \alpha ^{m}=\xi $ for some $m\in \mathbb {N}\}$ is a maximal nilpotent subsemigroup of $\mathcal {O}_{n}$ with zero $\xi $. Moreover, for a nonempty subset $Y$ of $X_{n}$, we give a necessary and sufficient condition for the set $\mathcal {O}_{n}(Y)$ to be a subsemigroup. Then we find a unique minimal generating set, and so rank, of $\mathcal {O}_{n}(Y)$ whenever it is a subsemigroup of $\mathcal {O}_{n}$. Every subset $Y$ of $X_{n}$ such that $\mathcal {O}_{n}(Y)$ is (completely) isolated was characterized.
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