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Associative, Idempotent, Symmetric, and Order-preserving Operations on Chains

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Abstract

We characterize the associative, idempotent, symmetric, and order-preserving binary operations on (finite) chains in terms of properties of (the Hasse diagram of) their associated semilattice order. In particular, we prove that the number of associative, idempotent, symmetric, and order-preserving operations on an n-element chain is the nth Catalan number.

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Acknowledgments

We gratefully acknowledge the anonymous referee for pointing out the connections between ≤n-preserving semilattice orders and C-posets (see Remark 4.15). We also thank Jean-Luc Marichal for his useful comments on the first version of the paper.

The first author is supported by the Luxembourg National Research Fund under the project PRIDE 15/10949314/GSM.

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  1. Mathematics Research Unit, FSTC, University of Luxembourg, 6 avenue de la Fonte, L-4364, Esch-sur-Alzette, Luxembourg

    Jimmy Devillet & Bruno Teheux

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  1. Jimmy Devillet
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  2. Bruno Teheux
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Correspondence to Jimmy Devillet.

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Devillet, J., Teheux, B. Associative, Idempotent, Symmetric, and Order-preserving Operations on Chains. Order 37, 45–58 (2020). https://doi.org/10.1007/s11083-019-09490-7

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