Abstract
We investigate monotone idempotent n-ary semigroups and provide a generalization of the Czogala–Drewniak Theorem, which describes the idempotent monotone associative functions having a neutral element. We also present a complete characterization of idempotent monotone n-associative functions on an interval that have neutral elements.
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Notes
If I is bounded and we denote the end-points of I by m and M (\(m\ge M\)), then \(\bar{I}=[m,M]\). If I is not bounded from below (or above), then we let \(m=-\infty \) (\(M=+\infty \), respectively). For instance, \(\bar{\mathbb {R}}=\mathbb {R}\cup \{\pm \infty \}\).
Let m and M be the boundary points of \(\bar{I}\). We use the convention that \(g(m-0)=M\) and \(g(M+0)=m\).
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Acknowledgements
The authors would like to express their gratitude to the referee, whose comments and suggestions highly improved the presentation of this article. One of the most important observations of the referee was that most of the results of Sect. 3 can be stated in a more general setting. The authors appreciate a lot the valuable suggestions of Mikhail Volkov about the final form of the paper.
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Communicated by Mikhail Volkov.
Supported by the project R-STR-1041-00-Z of the University of Luxembourg. Gergely Kiss was supported by the Hungarian Scientific Research Fund (OTKA) K104178 and the internal research project R-AGR-0500-MR03 of the University of Luxembourg. Gábor Somlai was supported by the Hungarian Scientific Research Fund (OTKA) K115799.
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Kiss, G., Somlai, G. A characterization of n-associative, monotone, idempotent functions on an interval that have neutral elements. Semigroup Forum 96, 438–451 (2018). https://doi.org/10.1007/s00233-017-9876-3
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DOI: https://doi.org/10.1007/s00233-017-9876-3