Algebra universalis

, 80:10 | Cite as

The core label order of a congruence-uniform lattice

  • Henri MühleEmail author


We investigate the alternate order on a congruence-uniform lattice \({\mathcal {L}}\) as introduced by N. Reading, which we dub the core label order of \({\mathcal {L}}\). When \({\mathcal {L}}\) can be realized as a poset of regions of a simplicial hyperplane arrangement, the core label order is always a lattice. For general \({\mathcal {L}}\), however, this fails. We provide an equivalent characterization for the core label order to be a lattice. As a consequence we show that the property of the core label order being a lattice is inherited to lattice quotients. We use the core label order to characterize the congruence-uniform lattices that are Boolean lattices, and we investigate the connection between congruence-uniform lattices whose core label orders are lattices and congruence-uniform lattices of biclosed sets.


Congruence-uniform lattices Interval doubling Semidistributive lattices Crosscut theorem Möbius function Biclosed sets 

Mathematics Subject Classification

06B05 06A07 



  1. 1.
    Bancroft, E.: Shard Intersections and Cambrian Congruence Classes in Type \(A\). Ph.D. thesis, North Carolina State University (2011)Google Scholar
  2. 2.
    Bancroft, E.: The shard intersection order on permutations (2011). arXiv:1103.1910
  3. 3.
    Barnard, E.: The canonical join complex (2016). arXiv:1610.05137
  4. 4.
    Björner, A., Edelman, P.H., Ziegler, G.M.: Hyperplane arrangements with a lattice of regions. Discrete Comput. Geom. 5, 263–288 (1990)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Clifton, A., Dillery, P., Garver, A.: The canonical join complex for biclosed sets. Algebra Universalis 79, 84 (2018)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Day, A.: Characterizations of finite lattices that are bounded-homomorphic images or sublattices of free lattices. Can. J. Math. 31, 69–78 (1979)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Day, A.: Doubling constructions in lattice theory. Can. J. Math. 44, 252–269 (1992)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Edelman, P.H.: A partial order on the regions of \({\mathbb{R}}^{n}\) dissected by hyperplanes. Trans. Am. Math. Soc. 283, 617–631 (1984)zbMATHGoogle Scholar
  9. 9.
    Freese, R., Ježek, J., Nation, J.B.: Free Lattices. American Mathematical Society, Providence (1995)CrossRefGoogle Scholar
  10. 10.
    Funayama, N., Nakayama, T.: On the distributivity of a lattice of lattice congruences. Proc. Imp. Acad. Tokyo 18, 553–554 (1942)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Garver, A., McConville, T.: Enumerative properties of grid-associahedra (2017). arXiv:1705.04901
  12. 12.
    Garver, A., McConville, T.: Oriented flip graphs of polygonal subdivisions and noncrossing tree partitions. J. Combin. Theory Ser. A 158, 126–175 (2018)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Grätzer, G.: Lattice Theory: Foundation. Springer, Basel (2011)CrossRefGoogle Scholar
  14. 14.
    McConville, T.: Biclosed Sets in Combinatorics. Ph.D. thesis, University of Minnesota (2015)Google Scholar
  15. 15.
    McConville, T.: Crosscut-simplicial lattices. Order 34, 465–477 (2017)MathSciNetCrossRefGoogle Scholar
  16. 16.
    McConville, T.: Lattice structure of grid-Tamari orders. J. Combin. Theory Ser. A 148, 27–56 (2017)MathSciNetCrossRefGoogle Scholar
  17. 17.
    McKenzie, R.: Equational bases and nonmodular lattice varieties. Trans. Am. Math. Soc. 174, 1–43 (1972)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Nation, J.B.: Unbounded semidistributive lattices. Algebra Logic 39, 87–92 (2000)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Petersen, T.K.: On the shard intersection order of a Coxeter group. SIAM J. Discrete Math. 27, 1880–1912 (2013)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Reading, N.: Lattice and order properties of the poset of regions in a hyperplane arrangement. Algebra Universalis 50, 179–205 (2003)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Reading, N.: The order dimension of the poset of regions in a hyperplane arrangement. J. Combin. Theory Ser. A 104, 265–285 (2003)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Reading, N.: Clusters, Coxeter-sortable elements and noncrossing partitions. Trans. Am. Math. Soc. 359, 5931–5958 (2007)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Reading, N.: Sortable elements and cambrian lattices. Algebra Universalis 56, 411–437 (2007)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Reading, N.: Noncrossing partitions and the shard intersection order. J. Algebr. Combin. 33, 483–530 (2011)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Reading, N.: Noncrossing arc diagrams and canonical join representations. SIAM J. Discrete Math. 29, 736–750 (2015)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Reading, N.: Lattice theory of the poset of regions. In: Grätzer, G., Wehrung, F. (eds.) Lattice Theory: Selected Topics and Applications, vol. 2, pp. 399–487. Birkhäuser, Cham (2016)CrossRefGoogle Scholar
  27. 27.
    Rota, G.C.: On the foundations of combinatorial theory I: theory of Möbius functions. Z. Wahrscheinlichkeitstheorie Verw. Gebiete 2, 340–368 (1964)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Sloane, N.J.A.: The online encyclopedia of integer sequences.
  29. 29.
    Stanley, R.P.: Enumerative Combinatorics, vol. 2. Cambridge University Press, Cambridge (2001)zbMATHGoogle Scholar
  30. 30.
    The Sage-Combinat Community: sage-combinat: enhancing Sage as a toolbox for computer exploration in algebraic combinatorics (2018).
  31. 31.
    The Sage Developers: Sage mathematics software system (version 8.5) (2018).

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institut für AlgebraTechnische Universität DresdenDresdenGermany

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