Abstract
Let \(\mathcal {POPE}_n\) be the monoid of all orientation-preserving and extensive partial transformations on \({\textbf {n}}=\{1,\dots , n\}\). In this paper, we characterize the structure of the generating sets of \(\mathcal {POPE}_n\), and prove that each generating set of \(\mathcal {POPE}_n\) contains a minimal idempotent generating set of \(\mathcal {POPE}_n\). Moreover, the minimal generating sets and minimal idempotent generating sets of \(\mathcal {POPE}_n\) coincide. As applications, we compute the number of distinct minimal (idempotent) generating sets of \(\mathcal {POPE}_n\), and prove that both the rank and the idempotent rank of the monoid \(\mathcal {POPE}_n\) are equal to \(\frac{n^2+n+2}{2}\). Finally, we determine the maximal subsemigroups as well as the maximal idempotent generated subsemigroups of the monoid \(\mathcal {POPE}_n\).
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The authors would like to thank the referee for his/her valuable suggestions and comments which help to improve the presentation of this paper.
Funding
This work was supported by the National Natural Science Foundation of China (No.12261022) and the National Natural Science Foundation of China (No.11461014).
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Communicated by Mikhail Volkov.
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Zhao, P., Hu, H. The monoid of all orientation-preserving and extensive partial transformations on a finite chain. Semigroup Forum 106, 720–746 (2023). https://doi.org/10.1007/s00233-023-10359-w
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DOI: https://doi.org/10.1007/s00233-023-10359-w