DM-GAA

ISSN 1509-9415 (print version)

ISSN 2084-0373 (electronic version)

https://doi.org/10.7151/dmgaa

Discussiones Mathematicae - General Algebra and Applications

Cite Score: 0.4

SJR: 0.203

SNIP: 0.562

MCQ: 0.12

Index Copernicus: 121.02

Discussiones Mathematicae - General Algebra and Applications

Article in volume


Authors:

G. Grätzer

George Grätzer

Department of Mathematics
University of Manitoba
Winnipeg, MB R3T 2N2, Canada

email: gratzer@me.com

Title:

Using the Swing Lemma and $\boldsymbol{\mathcal{C}_1}$-diagrams for congruences of planar semimodular lattices

PDF

Source:

Discussiones Mathematicae - General Algebra and Applications 43(1) (2023) 63-74

Received: 2021-06-06 , Revised: 2021-07-18 , Accepted: 2021-07-19 , Available online: 2023-01-11 , https://doi.org/10.7151/dmgaa.1410

Abstract:

A planar semimodular lattice $K$ is slim if $\mathsf{M}_{3}$ is not a sublattice of $K$. In a recent paper, G. Czédli found four new properties of congruence lattices of slim, planar, semimodular lattices, including the No Child Property: Let $\mathcal{P}$ be the ordered set of join-irreducible congruences of $K$. Let $x,y,z \in \mathcal{P}$ and let $z$ be a maximal element of $\mathcal{P}$. If $x \neq y$ and $x, y \prec z$ in $\mathcal{P}$, then there is no element $u$ of $\mathcal{P}$ such that $u \prec x, y$ in $\mathcal{P}$. The Swing Lemma and a standardized diagram type are used to give direct proofs of Czédli's four properties.

Keywords:

rectangular lattice, slim planar semimodular lattice, congruence lattice

References:

  1. G. Czédli, Patch extensions and trajectory colorings of slim rectangular lattices, Algebra Universalis 72 (2014) 125–154.
    https://doi.org/10.1007/s00012-014-0294-z
  2. G. Czédli, A note on congruence lattices of slim semimodular lattices, Algebra Universalis 72 (2014) 225–230.
    https://doi.org/10.1007/s00012-014-0286-z
  3. G. Czédli, Diagrams and rectangular extensions of planar semimodular lattices, Algebra Universalis 77 (2017) 443–498.
    https://doi.org/10.1007/s00012-017-0437-0
  4. G. Czédli, Lamps in slim rectangular planar semimodular lattices, Acta Sci. Math. (Szeged) 87 (2021) 381–413. \pagebreak
    https://doi.org/10.14232/actasm-021-865-y0
  5. G. Czédli, Non-finite axiomatizability of some finite structures. arXiv:2102.00526
  6. G. Czédli and G. Grätzer, Notes on planar semimodular lattices VII. Resections of planar semimodular lattices, Order 30 (2013) 847–858.
    https://doi.org/10.1007/s11083-012-9281-1
  7. G. Czédli and G. Grätzer, {Planar Semimodular Lattices: Structure and Diagrams}, Chapter 3 in \cite{LTS1}.
    https://doi.org/10.1007/978-3-319-06413-0_3
  8. G. Czédli and G. Grätzer, A new property of congruence lattices of slim, planar, semimodular lattices. arXiv:2103.04458
  9. G. Czédli and E.T. Schmidt, The Jordan-Hölder theorem with uniqueness for groups and semimodular lattices, Algebra Universalis 66 (1–2) (2011) 69–79.
    https://doi.org/10.1007/s00012-011-0144-1
  10. G. Czédli and E.T. Schmidt, Slim semimodular lattices I. A visual approach, ORDER 29 (2012) 481–497.
    https://doi.org/10.1007/s11083-011-9215-3
  11. A. Day, Characterizations of finite lattices that are bounded-homomorphic images or sublattices of free lattices, Canad. J. Math. 31 (1979) 69–78.
  12. R. Freese, J. Ježek and J.B. Nation, Free Lattices, Mathematical Surveys and Monographs 42 (American Mathematical Society, Providence, RI, 1995).
    https://doi.org/10.1090/surv/042
  13. G. Grätzer, {Planar Semimodular Lattices: Congruences}, Chapter 4 in \cite{LTS1}.
    https://doi.org/10.1007/978-3-319-06413-0\_4
  14. G. Grätzer, Notes on planar semimodular lattices VI. On the structure theorem of planar semimodular lattices, Algebra Universalis 69 (2013) 301–304.
    https://doi.org/10.1007/s00012-013-0233-4
  15. G. Grätzer, Two Topics Related to Congruence Lattices of Lattices, Chapter 10 in \cite{LTS1}.
    https://doi.org/10.1007/978-3-319-06413-0_10
  16. G. Grätzer, Congruences in slim, planar, semimodular lattices: The Swing Lemma, Acta Sci. Math. (Szeged) 81 (2015) 381–397.
    https://doi.org/10.1007/978-3-319-38798-7_25
  17. G. Grätzer, On a result of Gábor Czédli concerning congruence lattices of planar semimodular lattices, Acta Sci. Math. (Szeged) 81 (2015) 25–32.
    https://doi.org/10.14232/actasm-014-024-1
  18. G. Grätzer, The Congruences of a Finite Lattice, A {Proof-by-Picture} Approach, Second Edition (Birkhäuser, 2016). Part I is accessible at arXiv:2104.06539
    https://doi.org/10.1007/978-3-319-38798-7
  19. G. Grätzer, Congruences of fork extensions of lattices, Algebra Universalis 76 (2016) 139–154.
  20. G. Grätzer, Congruences and trajectories in planar semimodular lattices, Discuss. Math. GAA 38 (2018) 131–142.
    https://doi.org/10.7151/dmgaa.1280
  21. G. Grätzer, Notes on planar semimodular lattices VIII. Congruence lattices of SPS lattices, Algebra Universalis 81 (2020).
    https://doi.org/10.1007/s00012-020-0641-1
  22. G. Grätzer, A gentle introduction to congruences of planar semimodular lattices, Presentation at the meeting AAA 101, Novi Sad, 2021.
  23. G. Grätzer, Notes on planar semimodular lattices IX. On Czédli diagrams. arXiv:1307.0778
  24. G. Grätzer and E. Knapp, Notes on planar semimodular lattices I. Construction, Acta Sci. Math. (Szeged) 73 (2007) 445–462.
  25. G. Grätzer and E. Knapp, A note on planar semimodular lattices, Algebra Universalis 58 (2008) 497–499.
    https://doi.org/10.1007/s00012-008-2089-6
  26. G. Grätzer and E. Knapp, Notes on planar semimodular lattices II. Congruences, Acta Sci. Math. (Szeged) 74 (2008) 37–47.
  27. G. Grätzer and E. Knapp, Notes on planar semimodular lattices III. Rectangular lattices, Acta Sci. Math. (Szeged) 75 (2009) 29–48.
  28. G. Grätzer and E. Knapp, Notes on planar semimodular lattices IV. The size of a minimal congruence lattice representation with rectangular lattices, Acta Sci. Math. (Szeged) 76 (2010) 3–26.
  29. G. Grätzer, H. Lakser and E.T. Schmidt, Congruence lattices of finite semimodular lattices, Canad. Math. Bull. 41 (1998) 290–297.
    https://doi.org/10.4153/cmb-1998-041-7
  30. G. Grätzer and T. Wares, Notes on planar semimodular lattices V. Cover-preserving embeddings of finite semimodular lattices into simple semimodular lattices, Acta Sci. Math. (Szeged) 76 (2010) 27–33.
  31. G. Grätzer and F. Wehrung eds., Lattice Theory: Special Topics and Applications 1 (Birkhäuser Verlag, Basel, 2014).
    https://doi.org/10.1007/978-3-319-06413-0
  32. B. Jónsson and J.B. Nation, Representation of $2$-distributive modular lattices of finite length, Acta Sci. Math. (Szeged) 51 (1987) 123–128.
  33. R.N. McKenzie, Equational bases and nonmodular lattice varieties, Trans. Amer. Math. Soc. 174 (1972) 1–43.
  34. J.B. Nation, Bounded finite lattices. Universal algebra (Esztergom, 1977), 531–533, Colloq. Math. Soc. János Bolyai 29 (North-Holland, Amsterdam-New York, 1982).
  35. J.B. Nation, Revised Notes on Lattice Theory. http://www.math.hawaii.edu/ jb/books.html

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