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 (2023): 0.6

SJR (2023): 0.214

SNIP (2023): 0.604

Index Copernicus (2022): 121.02

H-Index: 5

Discussiones Mathematicae - General Algebra and Applications

Article in volume

Authors:

G. Czedli

Gabor Czedli

email: czedli@math.u-szeged.hu

Title:

Revisiting Faigle geometries from a perspective of semimodular lattices

PDF

Source:

Discussiones Mathematicae - General Algebra and Applications 43(2) (2023) 207-222

Received: 2021-07-22 , Revised: 2021-11-30 , Accepted: 2021-11-30 , Available online: 2023-01-11 , https://doi.org/10.7151/dmgaa.1416

Abstract:

In 1980, U. Faigle introduced a sort of finite geometries on posets that are in bijective correspondence with finite semimodular lattices. His result has almost been forgotten in lattice theory. Here we simplify the axiomatization of these geometries, which we call Faigle geometries. To exemplify their usefulness, we give a short proof of a theorem of Grätzer and E. Knapp (2009) asserting that each slim semimodular lattice $L$ has a congruence-preserving extension to a slim rectangular lattice of the same length as $L$. As another application of Faigle geometries, we give a short proof of G. Grätzer and E.W. Kiss' result from 1986 (also proved by M. Wild in 1993, the present author and E.T. Schmidt in 2010, and B. Skublics in 2013) that each finite semimodular lattice $L$ has an extension to a geometric lattice of the same length as $L$.

Keywords:

Faigle geometry, semimodular lattice, planar semimodular lattice, rectangular lattice, congruence-preserving extension, slim semimodular lattice, geometric lattice, cover-preserving extension

References:

  1. P. Crawley, R.P. Dilworth: Algebraic Theory of Lattices. Prentice Hall, 1973 ISBN: 0130222690, 9780130222695
  2. Czédli, G.: Representing homomorphisms of distributive lattices as restrictions of congruences of rectangular lattices, Algebra Universalis 67, 313–345 (2012) DOI: 10.1007/s00012-012-0190-3
  3. Czédli, G.: Slim patch lattices as absolute retracts and maximal lattices. http://arxiv.org/abs/2105.12868
  4. Czédli, G., Grätzer, G.: Planar semimodular lattices: structure and diagrams. Chapter 3 in: Grätzer, G., Wehrung, F. (eds.), Lattice Theory: Special Topics and Applications, pp 91–130, Birkhäuser, Basel (2014) DOI: 10.1007/978-3-319-06413-0\textunderscore 3
  5. Czédli, G., Kurusa, Á.: A convex combinatorial property of compact sets in the plane and its roots in lattice theory. Categories and General Algebraic Structures with Applications \tbf{11}, 57–92 (2019)\quad http://cgasa.sbu.ac.ir/article\textunderscore82639.html DOI: 10.29252/CGASA.11.1.57
  6. Czédli, G., Schmidt, E.T.: A cover-preserving embedding of semimodular lattices into geometric lattices. Advances in Mathematics \tbf{225}, 2455–2463 (2010) DOI: 10.1016/j.aim.2010.05.001
  7. Czédli, G., Schmidt, E.T.: Slim semimodular lattices. I. A visual approach. Order \tbf{29}, 481–497 (2012) DOI: 10.1007/s11083-011-9215-3
  8. Dilworth, R.P.: A decomposition theorem for partially ordered sets. Ann. of Math. \tbf{(2) 51}, 161–166 (1950) DOI:10.1007/978-0-8176-4842-8\textunderscore10
  9. Faigle, U.: Geometries on partially ordered sets. J. Combinatorial Theory B \tbf{28}, 26–51 (1980) DOI: 10.1016/0095-8956(80)90054-4
  10. Grätzer, G., Kiss, E. W.: A construction of semimodular lattices. Order \tbf{2}, 351–365 (1986) DOI: 10.1007/BF00367424
  11. Grätzer, G., Knapp, E.: Notes on planar semimodular lattices. I. Construction. Acta Sci.\ Math.\ (Szeged) \tbf{73}, 445–462 (2007)
  12. Grätzer, G., Knapp, E.: Notes on planar semimodular lattices. III. Congruences of rectangular lattices. Acta Sci. Math. (Szeged), \tbf{75}, 29–48 (2009)
  13. Grätzer, G., Schmidt, E. T.: The strong independence theorem for automorphism groups and congruence lattices of finite lattices. Beiträge Algebra Geom. \tbf{36}, 97–108 (1995)
  14. Kelly, D., Rival, I.: Planar lattices. Canadian J. Math. \tbf{27}, 636–665 (1975) DOI: 10.4153/CJM-1975-074-0
  15. Quackenbush, R. W.: Review on Faigle \cite{faigle}. MathSciNet, MR565509 (81m:05054)
  16. Skublics, B.: Isometrical embeddings of lattices into geometric lattices. Order 30, 797–806 (2013) DOI: 10.1007/s11083-012-9277-x
  17. Stern, M.: Semimodular Lattices –- Theory and Application, Cambridge Univ. Press, 1999 DOI: 10.1017/CBO9780511665578
  18. Wild, M.: Cover preserving embedding of modular lattices into partition lattices. Discrete Math. \tbf{112}, 207–244 (1993) DOI: 10.1016/0012-365X(93)90235-L
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