Article
| Title: | On semigroups of order-preserving transformations with the same fix set (English) |
| Author: | Ayık, Gonca |
| Author: | Ayık, Hayrullah |
| Author: | Koppitz, Jörg |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 76 |
| Issue: | 2 |
| Year: | 2026 |
| Pages: | 379-399 |
| Summary lang: | English |
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| Category: | math |
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| 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. (English) |
| Keyword: | order-preserving transformation |
| Keyword: | orientation-preserving |
| Keyword: | (completely) isolated subsemigroup |
| Keyword: | generating set |
| Keyword: | rank |
| MSC: | 20M05 |
| MSC: | 20M20 |
| DOI: | 10.21136/CMJ.2026.0504-24 |
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| Date available: | 2026-05-22T11:17:47Z |
| Last updated: | 2026-05-25 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/153640 |
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