Congruences of finite semidistributive lattices

J. B. Nation

doi:10.56754/0719-0646.2603.443

DOI:

https://doi.org/10.56754/0719-0646.2603.443

Abstract

We show that there are finite distributive lattices that are not the congruence lattice of any finite semidistributive lattice. For \(0 \leq k \leq 2\), the distributive lattice \((\mathbf{B}_k)_{++} = \mathbf{2} + \mathbf{B}_k\), where \(\mathbf{B}_k\) denotes the boolean lattice with \(k\) atoms, is not the congruence lattice of any finite semidistributive lattice. Neither can these lattices be a filter in the congruence lattice of a finite semidistributive lattice. However, each \((\mathbf{B}_k)_{++}\) with \(k \geq 3\) is the congruence lattice of a finite semidistributive lattice, say \(\mathbf{L}_k\). These lattices \(\mathbf{L}_k\) cannot be bounded (in the sense of McKenzie), as no \((\mathbf{B}_k)_{++}\) \((k \geq 0)\) is the congruence lattice of a finite bounded lattice. A companion paper shows that every \((\mathbf{B}_k)_{++}\) \((k \geq 0)\) can be represented as the congruence lattice of an infinite semidistributive lattice. We also find sufficient conditions for a finite distributive lattice to be representable as the congruence lattice of a finite bounded (and hence semidistributive) lattice.

Keywords

Distributive lattice , semidistributive lattice , congruence lattice

Mathematics Subject Classification:

06B10 , 06B15

J. B. Nation Department of Mathematics, University of Hawaii, Honolulu, HI 96822, USA. https://orcid.org/0000-0001-5959-3304

Pages: 443–473
Date Published: 2024-10-18
Vol. 26 No. 3 (2024)

K. Adaricheva, R. Freese, and J. Nation, “Notes on join semidistributive lattices,” Int. J. Algebra and Computation, vol. 32, pp. 347–356, 2020.

K.Adaricheva and J. Nation, “Bases of closure systems (Chapter6),” in Lattice Theory: Special Topics and Applications, G. Grätzer and F. Wehrung, Eds. Cham: Birkhäuser, 2016, vol. 2, pp. 181–213.

K. Adaricheva and J. Nation, “Classes of semidistributive lattices (Chapter 3),” in Lattice Theory: Special Topics and Applications, G. Grätzer and F. Wehrung, Eds. Cham: Birkhäuser, 2016, vol. 2, pp. 59–101.

K. Adaricheva, J. Nation, and R. Rand, “Ordered direct implicational basis of a finite closure system,” Discrete Applied Math., vol. 161, pp. 707–723, 2013.

A. Day, “Characterizations of finite lattices that are bounded-homomorphic images of sublattices of free lattices,” Canad. J. Math., vol. 31, pp. 69–78, 1979.

A. Day, “Doubling constructions in lattice theory,” Canad. J. Math., vol. 44, pp. 252–269, 1994.

R. Freese, J. Ježek, and J. Nation, Free Lattices, ser. Mathematical Surveys and Monographs. Providence: Amer. Math. Soc., 1995, vol. 42.

R. Freese and J. Nation, “A simple semidistributive lattice,” Int. J. Algebra and Computation, vol. 31, pp. 219–224, 2021.

W. Geyer, “The generalized doubling construction and formal concept analysis,” Algebra Universalis, vol. 32, pp. 341–367, 1994.

G. Grätzer, “A property of transferable lattices,” Proc. Amer. Math.Soc., vol. 43, pp. 269–271, 1974.

G. Grätzer and J. Nation, “Congruence lattices of infinite semidistributive lattices,” 2023, arXiv:2306.04113; extended version available at /math.hawaii.edu/∼jb/.

B. Jónsson, “Sublattices of a free lattice,” Canad. J. Math, vol. 13, pp. 256–264, 1961.

B. Jónsson and J. Kiefer, “Finite sublattices of a free lattice,” Canad. J. Math, vol. 14, pp. 487–497, 1962.

B. Jónsson and J. Nation, “A report on sublattices of a free lattice,” in Universal Algebra and Lattice Theory, Contributions to Universal Algebra, ser. Lecture Notes in Mathematics, Coll. Math. Soc. János Bolyai. North-Holland, 1977, vol. 17, pp. 223–257.

R. McKenzie, “Equational bases and non-modular lattice varieties,” Trans. Amer. Math. Soc., vol. 174, pp. 1–43, 1972.

J. Nation, “Finite sublattices of a free lattice,” Trans. Amer. Math. Soc., vol. 269, pp. 311–327, 1982.

J. Nation, “Notes on lattice theory,” 1990, available at /math.hawaii.edu/∼jb/.

J. Nation, “Unbounded semidistributive lattices,” Algebra and Logic (Russian), vol. 39, pp. 87–92, 2000.

P. Pudlák and J. Tůma, “Yeast graphs and fermentation of algebraic lattices,” in Lattice Theory (Szeged, 1974), Colloq. Math. Soc. János Bolyai. Amsterdam: North Holland, 1976, vol. 14, pp. 301–341.