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The fundamental theorem of finite semidistributive lattices

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Abstract

We prove a Fundamental Theorem of Finite Semidistributive Lattices (FTFSDL), modelled on Birkhoff’s Fundamental Theorem of Finite Distributive Lattices. Our FTFSDL is of the form “A poset L is a finite semidistributive lattice if and only if there exists a set with some additional structure, such that L is isomorphic to the admissible subsets of ordered by inclusion; in this case, and its additional structure are uniquely determined by L.” The additional structure on is a combinatorial abstraction of the notion of torsion pairs from representation theory and has geometric meaning in the case of posets of regions of hyperplane arrangements. We show how the FTFSDL clarifies many constructions in lattice theory, such as canonical join representations and passing to quotients, and how the semidistributive property interacts with other major classes of lattices. Many of our results also apply to infinite lattices.

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Acknowledgements

H.T. would like to thank Laurent Demonet and Osamu Iyama for their hospitality at Nagoya University and helpful conversations. D.E.S. would like to thank the attendees of the Maurice Auslander Distinguished Lectures in 2019 for their helpful remarks. All the authors would like to thank the anonymous referees for their careful reading and valuable suggestions.

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Authors and Affiliations

  1. Department of Mathematics, NC State University, Box 8205, Raleigh, NC, 27695-8205, USA

    Nathan Reading

  2. Department of Mathematics, University of Michigan, 2074 East Hall, 530 Church Street, Ann Arbor, MI, 48109-1043, USA

    David E Speyer

  3. Département de mathématiques, UQAM, C.P. 8888, Succursale Centre-ville, PK-5151, Montréal, QC, H3C 3P8, Canada

    Hugh Thomas

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  1. Nathan Reading
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  2. David E Speyer
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Correspondence to Nathan Reading.

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Nathan Reading was partially supported by the National Science Foundation under Grant Number DMS-1500949 and by the Simons Foundation under Award Number 581608. David Speyer was partially supported by the National Science Foundation under Grant Number DMS-1600223. Hugh Thomas was partially supported by an NSERC Discovery Grant and the Canada Research Chairs program.

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Reading, N., Speyer, D.E. & Thomas, H. The fundamental theorem of finite semidistributive lattices. Sel. Math. New Ser. 27, 59 (2021). https://doi.org/10.1007/s00029-021-00656-z

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