Abstract
We investigate classifications of quasitrivial semigroups defined by certain equivalence relations. The subclass of quasitrivial semigroups that preserve a given total ordering is also investigated. In the special case of finite semigroups, we address and solve several related enumeration problems.
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Notes
In fact, we have \(|{\mathcal {F}}_n| \sim \frac{1}{2\lambda +1}{\,}n!{\,}\lambda ^{n+2}\) as \(n\rightarrow \infty\), where \(\lambda \,(\approx 1.71)\) is the inverse of the unique positive zero of the real function \(x\mapsto x+3-2e^x\).
Recall that \(k=|X_n/{\sim }_F|\) and that \(n_i=|C_i|\) for \(i=1,\ldots ,k\).
In particular, we observe that, for any \(F\in {\mathcal {F}}_n\), the number of distinct values in the sequence \(|F^{-1}|\) is exactly \(|X_n/{\sim _F}|\).
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Acknowledgements
This research is supported by the Internal Research Project R-AGR-0500 of the University of Luxembourg and the Luxembourg National Research Fund PRIDE 15/10949314/GSM.
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Communicated by Mikhail Volkov.
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Devillet, J., Marichal, JL. & Teheux, B. Classifications of quasitrivial semigroups. Semigroup Forum 100, 743–764 (2020). https://doi.org/10.1007/s00233-020-10087-5
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DOI: https://doi.org/10.1007/s00233-020-10087-5