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A property of meets in slim semimodular lattices and its application to retracts

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Abstract

Slim semimodular lattices were introduced by G. Grätzer and E. Knapp in 2007, and they have intensively been studied since then. These lattices can be given by \(\mathcal{C}_1\)-diagrams, defined by the author in 2017. We prove that if x and y are incomparable elements in such a lattice L, then their meet has the property that the interval \([x \wedge y, x]\) is a chain, this chain is of a normal slope in every \(\mathcal{C}_1\)-diagram of L, and except possibly for x, the elements of this chain are meet-reducible.

In the direct square K1 of the three-element chain, let X1 and A1 be the set of atoms and the sublattice generated by 0 and the coatoms, respectively. Denote by K2 the unique eight-element lattice embeddable in K1. Let A2 be the sublattice of K2 consisting of 0, 1, the meet-reducible atom, and the join-reducible coatom. Let X2 stand for the singleton consisting of the doubly reducible element of K2. For i = 1, 2, we apply the above-mentioned property of meets to prove that whenever Ki is a sublattice and Si is a retract of a slim semimodular lattice, then \(A_i \subseteq S_i\) implies that \(X_i \subseteq S_i\).

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References

  1. L. Boyu, All retraction operators on a complete lattice form a complete lattice, Acta Mathematica Sinica, 7 (1991), 247–251.

  2. G. Czédli, Patch extensions and trajectory colorings of slim rectangular lattices, Algebra Universalis, 72 (2014), 125–154.

  3. G. Czédli, Diagrams and rectangular extensions of planar semimodular lattices, Algebra Universalis, 77 (2017), 443–498.

  4. G. Czédli, Lamps in slim rectangular planar semimodular lattices, Acta Sci. Math. (Szeged), 87 (2021), 381–413.

  5. G. Czédli, Slim patch lattices as absolute retracts and maximal lattices, arXiv (2021), http://arxiv.org/abs/2105.12868.

  6. G. Czédli, Revisiting Faigle geometries from a perspective of semimodular lattices (Extended version), arXiv (2021), http://arxiv.org/abs/2107.10202.

  7. G. Czédli, Lattices of retracts of direct products of two finite chains and notes on retracts of lattices, Algebra Universalis, 83 (2022), 34.

  8. G. Czédli and G. Grätzer, Planar semimodular lattices: structure and diagrams. Chapter 3, Lattice Theory: Special Topics and Applications, Eds.: G. Grätzer, F. Wehrung, Birkhäuser, Basel, 2014, pp. 91–130.

  9. G. Czédli and Á. Kurusa, A convex combinatorial property of compact sets in the plane and its roots in lattice theory, Categories and General Algebraic Structures with Applications, 11 (2019), 57–92; http://cgasa.sbu.ac.ir/article_82639.html.

  10. G. Czédli and A. Molkhasi, Absolute retracts for finite distributive lattices and slim semimodular lattices, Order, doi: 10.1007/s11083-021-09592-1.

  11. G. Czédli and E.T. Schmidt, The Jordan–Hölder theorem with uniqueness for groups and semimodular lattices, Algebra Universalis, 66 (2011), 69–79.

  12. G. Czédli and E.T. Schmidt, Slim semimodular lattices. I. A visual approach, Order, 29 (2012), 481–497.

  13. G. Grätzer, Lattice Theory: Foundation, Birkhäuser, Basel, 2011.

  14. G. Grätzer, The Congruences of a Finite Lattice, a Proof-by-Picture Approach, second edition, Birkhäuser, 2016.

  15. G. Grätzer, Notes on planar semimodular lattices. IX. C1-diagrams, Discussiones Mathematicae — General Algebra and Applications, submitted; https://www.researchgate.net/publication/350788941.

  16. G. Grätzer and E. Knapp, Notes on planar semimodular lattices. I. Construction, Acta Sci. Math. (Szeged), 73 (2007), 445–462.

  17. D. Jakubíková–Studenovská and J. Pócs, Lattice of retracts of monounary algebras, Math. Slovaca, 61 (2011), 107–125.

  18. D. Kelly and I. Rival, Planar lattices, Canadian J. Math., 27 (1975), 636– 665.

  19. J. B. Nation, Notes on Lattice Theory, http://www.math.hawaii.edu/~jb/math618/Nation-LatticeTheory.pdf.

  20. I. Rival, The retract construction, Ordered sets (Banff, Alta., 1981), NATO Adv. Study Inst. Ser. C: Math. Phys. Sci., 83, Reidel, Dordrecht–Boston, Mass., 1982, pp.97–122.

  21. L. Zádori, Series parallel posets with nonfinitely generated clones, Order, 10 (1993), 305–316.

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Authors and Affiliations

  1. Bolyai Institute, University of Wisconsin-La Crosse, 1725 State St, WI, 54601, Szeged, Hungary

    Gábor Czédli

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  1. Gábor Czédli
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Correspondence to Gábor Czédli.

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Dedicated to Professor László Stachó on the occasion that he received the Béla Szőkefalvi-Nagy Medal of Acta Scientiarum Mathematicarum for the year 2021

This research was supported by the National Research, Development and Innovation Fund of Hungary under funding scheme K 134851.

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Czédli, G. A property of meets in slim semimodular lattices and its application to retracts. Acta Sci. Math. (Szeged) 88, 595–610 (2022). https://doi.org/10.1007/s44146-022-00040-z

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  • DOI: https://doi.org/10.1007/s44146-022-00040-z

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