Abstract
In this article we investigate the lattices of Dyck paths of type A and B under dominance order, and explicitly describe their Heyting algebra structure. This means that each Dyck path of either type has a relative pseudocomplement with respect to some other Dyck path of the same type. While the proof that this lattice forms a Heyting algebra is quite straightforward, the explicit computation of the relative pseudocomplements using the lattice-theoretic definition is quite tedious. We give a combinatorial description of the Heyting algebra operations join, meet, and relative pseudocomplement in terms of height sequences, and we use these results to derive formulas for pseudocomplements and to characterize the regular elements in these lattices.
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This work was funded by the FWF Research Grant No. Z130-N13, as well as a Public Grant overseen by the French National Research Agency (ANR) as part of the “Investissements d’Avenir” Program (Reference: ANR-10-LABX-0098), and Digiteo project PAAGT (Nr. 2015-3161D).
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Mühle, H. A Heyting Algebra on Dyck Paths of Type A and B . Order 34, 327–348 (2017). https://doi.org/10.1007/s11083-016-9403-2
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DOI: https://doi.org/10.1007/s11083-016-9403-2