Accumulation points of congruence densities of finite lattices: Dedicated to George Grätzer on the occasion of his ninetieth birthday, with appreciation and admiration, and in friendship

Gábor Czédli

doi:10.56754/0719-0646.2803.455

Authors

Gábor Czédli University of Szeged, Bolyai Institute, Szeged, Aradi vértanúk tere 1, Hungary

DOI:

https://doi.org/10.56754/0719-0646.2803.455

Keywords:

Congruence density, accumulation point, lattice congruence, dually well-ordered set, semimodular lattice, modular lattice

Abstract

Let \(\mathcal{W}\) be a non-trivial variety of lattices and let \(L\) be a finite lattice in \(\mathcal{W}\). The congruence density of \(L\) with respect to \(\mathcal{W}\) is the number of congruences of \(L\) divided by the maximum number of congruences of \(|L|\)-element lattices belonging to \(\mathcal{W}\). We prove that, with respect to the order and multiplication of the real numbers, the set \(\mathrm{SCD}(\mathcal{W})\) of congruence densities of finite members of \(\mathcal{W}\) as well as its topological closure are countably infinite dually well-ordered monoids. We also prove that the set of accumulation points of \(\mathrm{SCD}(\mathcal{W})\) is either a singleton or it is countably infinite; furthermore, it is a singleton if and only if \(\mathcal{W}\) is a subvariety of the variety of modular lattices. This gives a complicated characterization of modularity: a non-singleton lattice \(K\) is modular if and only if \(\mathrm{SCD}(\mathcal{V}(K))\), where \(\mathcal{V}(K)\) denotes the variety generated by \(K\), has only one accumulation point. The class \(\mathcal{S}\) of semimodular lattices is not a variety, but \(\mathrm{SCD}(\mathcal{S})\) is still meaningful. We prove that \(\mathrm{SCD}(\mathcal{S})\) has exactly one accumulation point.

Downloads

Download data is not yet available.

References

[1] K. Adaricheva, R. Freese, and J. B. Nation, "Notes on join semidistributive lattices," Int. J. Algebra Comput., vol. 32, no. 2, pp. 347–356, 2022, doi: 10.1142/S0218196722500175.

[2] D. Ahmed and E. K. Horvath, "Yet two additional large numbers of subuniverses of finite lattices," Discuss. Math., Gen. Algebra Appl., vol. 39, no. 2, pp. 251–261, 2019, doi: 10.7151/dmgaa.1309.

[3] D. Ahmed, M. Salih, and D. Haje, "Analyzing Subuniverse Counts in Finite Semilattices: Unveiling the Rankings and Descriptions," 2024, arXiv.2408.09595.

[4] G. Czédli, "A note on finite lattices with many congruences," Acta Univ. M. Belii, Ser. Math., vol. 26, pp. 19–25, 2018.

[5] G. Czédli, "Eighty-three sublattices and planarity," Algebra Univers., vol. 80, no. 4, 2019, Art. ID 45, doi: 10.1007/s00012-019-0615-3.

[6] G. Czédli, "Finite semilattices with many congruences," Order, vol. 36, no. 2, pp. 233–247, 2019, doi: 10.1007/s11083-018-9464-5.

[7] G. Czédli, "Lattices with many congruences are planar," Algebra Univers., vol. 80, no. 1, 2019, Art. ID 16, doi: 10.1007/s00012-019-0589-1.

[8] G. Czédli, "Planar semilattices and nearlattices with eighty-three subnearlattices," Acta Sci. Math., vol. 86, pp. 117–165, 2020, doi: 10.14232/actasm-019-573-4.

[9] G. Czédli, "Revisiting Faigle geometries from a perspective of semimodular lattices," 2021, arXiv.2107.10202.

[10] G. Czédli, "The largest and all subsequent numbers of congruences of n-element lattices," Ural Math. J., vol. 11, no. 2, pp. 81–99, 2025, doi: 10.15826/umj.2025.2.006.

[11] G. Czédli, "Notes on the congruence densities and quasiorder densities of sublattices," in Algebra and Model Theory: Collection of Papers from the 17th Summer School “Problems Allied to Model Theory and Universal Algebra”, Erlagol–Novosibirsk, 2025, pp. 31–44.

[12] G. Czédli, "Lattices with congruence densities larger than 3/32," 2026, arXiv.2602.04321.

[13] R. Freese, "Computing congruence lattices of finite lattices," Proc. Am. Math. Soc., vol. 125, pp. 3457–3463, 1997, doi: 10.1090/S0002-9939-97-04332-3.

[14] G. Grätzer, "Congruences and prime-perspectivities in finite lattices." Algebra Univers., vol. 74, no. 3-4, pp. 351–359, 2015, doi: 10.1007/s00012-015-0355-y.

[15] G. Grätzer, Lattice Theory: Foundation, 1st ed. Basel: Birkhäuser, 2011, doi: 10.1007/978-3-0348-0018-1.

[16] G. Grätzer, "Planar semimodular lattices: congruences," in Lattice theory: special topics and applications. Volume 1. Cham: Birkhäuser/Springer, 2014, pp. 131–165, doi: 10.1007/978-3-319-06413-0_4.

[17] G. Grätzer, The congruences of a finite lattice. A “proof-by-picture” approach, 3rd ed. Cham: Birkhäuser, 2023, doi: 10.1007/978-3-031-29063-3.

[18] L. Kwuida and C. Mureşan, "On nontrivial weak dicomplementations and the lattice congruences that preserve them," Order, vol. 40, no. 2, pp. 423–453, 2023, doi: 10.1007/s11083-022-09597-4.

[19] C. Mureşan and J. Kulin, "On the largest numbers of congruences of finite lattices," Order, vol. 37, no. 3, pp. 445–460, 2020, doi: 10.1007/s11083-019-09514-2.

[20] F. P. Ramsey, "On a problem of formal logic." Proc. Lond. Math. Soc. (2), vol. 30, pp. 264–286, 1929, doi: 10.1112/plms/s2-30.1.264.

[21] N. E. Zaja, D. Haje, and D. Ahmed, "The sixth and seventh largest number of subuniverses of finite lattices," Academic Journal of Nawroz University, vol. 12, no. 1, pp. 143–148, 2023, doi: 10.25007/ajnu.v12n1a1713.