Abstract
We investigate the standard context, denoted by \(\mathbb {K}\left( \mathcal {L}_{n}\right) \), of the lattice \(\mathcal {L}_{n}\) of partitions of a positive integer n under the dominance order. Motivated by the discrete dynamical model to study integer partitions by Latapy and Duong Phan and by the characterization of the supremum and (infimum) irreducible partitions of n by Brylawski, we show how to construct the join-irreducible elements of \(\mathcal {L}_{n+1}\) from \(\mathcal {L}_{n}\). We employ this construction to count the number of join-irreducible elements of \(\mathcal {L}_{n}\), and confirm that the number of objects (and attributes) of \(\mathbb {K}\left( \mathcal {L}_{n}\right) \) has order \(\varTheta (n^2)\). We also discuss the embeddability of \(\mathbb {K}\left( \mathcal {L}_{n}\right) \) into \(\mathbb {K}\left( \mathcal {L}_{n+1}\right) \) with special emphasis on \(n=9\).
The third author gratefully acknowledges financial support by the Asociación Mexicana de Cultura A.C.
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Acknowledgements
The authors are grateful to Prof. Bernhard Ganter for pointing out the topic, for helpful advice and encouraging remarks. They also would like to thank Dr. Christian Meschke for his constant support. Moreover, they appreciate the constructive comments and suggestions given by the anonymous referees, which improved the presentation of the material.
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Behrisch, M., Chavarri Villarello, A., Vargas-García, E. (2021). Representing Partition Lattices Through FCA. In: Braud, A., Buzmakov, A., Hanika, T., Le Ber, F. (eds) Formal Concept Analysis. ICFCA 2021. Lecture Notes in Computer Science(), vol 12733. Springer, Cham. https://doi.org/10.1007/978-3-030-77867-5_1
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