Abstract
We characterize the canonical bases of lattices which attain the upper bound in Sauer-Shelah’s lemma, i.e., the (n, k)-extremal lattices of [AC15]. A characteristic construction of such bases is presented. We make the case that this approach sheds light on important combinatorial properties. In particular, we give an explicit description of an (n, k)-extremal lattice with precisely \({n \atopwithdelims ()k-1} +k-2\) meet-irreducibles, together with its canonical basis and Whitney numbers.
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- 1.
The restriction of having at most (and not exactly) n join-irreducibles is technical: the advantage is that (n, 1)-extremal lattices exist for any n. It makes no difference for the conclusions we draw.
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Albano, A. (2017). The Implication Logic of (n, k)-Extremal Lattices. In: Bertet, K., Borchmann, D., Cellier, P., Ferré, S. (eds) Formal Concept Analysis. ICFCA 2017. Lecture Notes in Computer Science(), vol 10308. Springer, Cham. https://doi.org/10.1007/978-3-319-59271-8_3
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DOI: https://doi.org/10.1007/978-3-319-59271-8_3
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