Geodesic
Using the Swing Lemma and \( \mathcal{C}_1 \)-diagrams for congruences of planar semimodular lattices
Discussiones Mathematicae. General Algebra and Applications, Tome 43 (2023) no. 1, pp. 63-74.

Voir la notice de l'article provenant de la source Library of Science

A planar semimodular lattice K is slim if 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 𝒫 be the ordered set of join-irreducible congruences of K. Let x,y,z ∈𝒫 and let z be a maximal element of 𝒫. If x y and x, y ≺ z in 𝒫, then there is no element u of 𝒫 such that u ≺ x, y in 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 
@article{DMGAA_2023_43_1_a5,
     author = {Gr\"atzer, George},
     title = {Using the {Swing} {Lemma} and \( {\mathcal{C}_1} \)-diagrams for congruences of planar semimodular lattices},
     journal = {Discussiones Mathematicae. General Algebra and Applications},
     pages = {63--74},
     publisher = {mathdoc},
     volume = {43},
     number = {1},
     year = {2023},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGAA_2023_43_1_a5/}
}
TY  - JOUR
AU  - Grätzer, George
TI  - Using the Swing Lemma and \( \mathcal{C}_1 \)-diagrams for congruences of planar semimodular lattices
JO  - Discussiones Mathematicae. General Algebra and Applications
PY  - 2023
SP  - 63
EP  - 74
VL  - 43
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMGAA_2023_43_1_a5/
LA  - en
ID  - DMGAA_2023_43_1_a5
ER  - 
%0 Journal Article
%A Grätzer, George
%T Using the Swing Lemma and \( \mathcal{C}_1 \)-diagrams for congruences of planar semimodular lattices
%J Discussiones Mathematicae. General Algebra and Applications
%D 2023
%P 63-74
%V 43
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMGAA_2023_43_1_a5/
%G en
%F DMGAA_2023_43_1_a5
Grätzer, George. Using the Swing Lemma and \( \mathcal{C}_1 \)-diagrams for congruences of planar semimodular lattices. Discussiones Mathematicae. General Algebra and Applications, Tome 43 (2023) no. 1, pp. 63-74. http://geodesic.mathdoc.fr/item/DMGAA_2023_43_1_a5/

[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