Abstract
Let L be an n-element finite lattice. We prove that if L has more than \(2^{n-5}\) congruences, then L is planar. This result is sharp, since for each natural number \(n\ge 8\), there exists a non-planar lattice with exactly \(2^{n-5}\) congruences.
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Adaricheva, K., Czédli, G.: Note on the description of join-distributive lattices by permutations. Algebra Univ. 72, 155–162 (2014)
Crawley, P., Dilworth, R.P.: Algebraic Theory of Lattices. Prentice, Englewood Cliffs (1973)
Czédli, G.: A note on finite lattices with many congruences. Acta Universitatis Matthiae Belii Ser. Math. Online, pp. 22–28. http://actamath.savbb.sk/pdf/oacta2018003.pdf (2018)
Dilworth, R.P.: The structure of relatively complemented lattices. Ann. Math. 51, 348–359 (1950)
Freese, R.: Computing congruence lattices of finite lattices. Proc. Am. Math. Soc. 125, 3457–3463 (1997)
Freese, R., Ježek, J., Nation, J.B.: Free lattices. In: Mathematical Surveys and Monographs, vol. 42. American Mathematical Society, Providence (1995)
Grätzer, G.: Lattice Theory: Foundation. Birkhäuser, Basel (2011)
Grätzer, G.: Congruences and prime-perspectivities in finite lattices. Algebra Univ. 74, 351–359 (2015)
Kelly, D., Rival, I.: Planar lattices. Can. J. Math. 27, 636–665 (1975)
Kulin, J., Mureşan, C.: Some extremal values of the number of congruences of a finite lattice. arxiv:1801.05282 (2018)
Mureşan, C.: Cancelling congruences of lattices while keeping their filters and ideals. arxiv:1710.10183 (2017)
Nation, J.B.: Notes on Lattice Theory. http://www.math.hawaii.edu/~jb/books.html
Rival, I.: Lattices with doubly irreducible elements. Can. Math. Bull. 17, 91–95 (1974)
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Dedicated to the memory of Ivan Rival.
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This research was supported by the National Research, Development and Innovation Fund of Hungary under the KH 126581 funding scheme.
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Czédli, G. Lattices with many congruences are planar. Algebra Univers. 80, 16 (2019). https://doi.org/10.1007/s00012-019-0589-1
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DOI: https://doi.org/10.1007/s00012-019-0589-1