Conjecture on Maximal Sublattices of Finite Semidistributive Lattices and Beyond

K. Adaricheva; A. Mata; S. Silberger; A. Zamojska-Dzienio

doi:10.46298/dmtcs.16374

Article

Conjecture on Maximal Sublattices of Finite Semidistributive Lattices and Beyond

K. Adaricheva⁽¹⁾, A. Mata⁽²⁾, S. Silberger⁽¹⁾, A. Zamojska-Dzienio⁽²⁾

(1) Hofstra University

(2) Warsaw University of Technology [WUT]

Download article

Open on arXiv

Submitted on

August 20, 2025

Accepted on

May 19, 2026

Published on

July 23, 2026

Last modified on

July 23, 2026

Volume 28

vol. 28:2

Combinatorics

DOI

10.46298/dmtcs.16374

License

Attribution 4.0 International (CC BY 4.0)

Indicators

163

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Conjecture on Maximal Sublattices of Finite Semidistributive Lattices and Beyond

K. Adaricheva⁽¹⁾, A. Mata⁽²⁾, S. Silberger⁽¹⁾, A. Zamojska-Dzienio⁽²⁾

(1) Hofstra University

(2) Warsaw University of Technology [WUT]

Abstract

We study maximal sublattices of finite semidistributive lattices via their complements. We focus on the conjecture that such complements are always intervals, which is known to be true for bounded lattices. Since the class of semidistributive lattices is the intersection of classes of join- and meet-semidistributive lattices, we study also complements for these classes, and in particular convex geometries of convex dimension 2, which is a subclass of join-semidistributive lattices. In the latter case, we describe the complements of maximal sublattices completely, as well as the procedure of finding all complements of maximal sublattices. 21 pages, 11 figures

Keywords

Rings and Algebras
06A07, 06A15, 52A37, 52C05, 52C07

Mathematics Subject Classification 2020

06A07 - Combinatorics of partially ordered sets
06A15 - Galois correspondences, closure operators (in relation to ordered sets)
52A37 - Other problems of combinatorial convexity
52C05 - Lattices and convex bodies in \(2\) dimensions (aspects of discrete geometry)
52C07 - Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)

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References

M. E. Adams, The Frattini sublattice of a distributive lattice, Algebra Universalis 3 (1973), 216-228.
DOI : 10.1007/bf02945121
M. E. Adams, R. Freese, J. B. Nation, J. Schmid, Maximal Sublattices and Frattini Sublattices of Bounded Lattices, J. Austral. Math. Soc., A 63 (1997), 110-127.
DOI : 10.1017/s1446788700000355
K. Adaricheva, J.B. Nation, Convex geometries, in Lattice Theory: Special Topics and Applications, v.2, G. Grätzer and F. Wehrung, eds. Springer, Basel, 2016.
DOI : 10.1007/978-3-319-44236-5_5
K. Adaricheva et al., Convex geometries representable by at most five circles on the plane, In-volve, a Journal of Mathematics 17 (2024), 337-354. Appendices (433 pages) to be found at https://arxiv.org/abs/2008.13077.
DOI : 10.2140/involve.2024.17.337
G. Czédli, Á. Kurusa, A convex combinatorial property of compact sets in the plane and its roots in lattice theory, Categ. Gen. Algebr. Struct. Appl. 11 (2019), 57-92.
DOI : 10.29252/cgasa.11.1.57
C.C. Chen, K.M. Koh, S.K. Tan, Frattini sublattices of distributive lattices, Algebra Universalis 3 (1973), 294-303.
DOI : 10.1007/bf02945131
C.C. Chen, K.M. Koh, S.K. Tan, On the Frattini sublattice of a finite distributive lattice, Algebra Universalis 5 (1975), 88-97.
DOI : 10.1007/bf02485236
A. Day, Doubling construction in lattice theory, Canad. J. Math. 44 (1992), 252-269.
DOI : 10.4153/cjm-1992-017-7
P. H. Edelman, R. E. Jamison, The theory of convex geometries, Geom. Dedicata 19 (1985), 247-270.
DOI : 10.1007/bf00149365
R. Freese, J. Ježek, J.B. Nation, Free Lattices, AMS, 1995.
DOI : 10.1007/bf00403408
B. Ganter, "Properties of finite lattices" by S. Reeg and W. Weiss, revisited, ICFCA 2019, LNAI 11511, D. Cristea et al. (Eds), 99-109.
DOI : 10.1007/978-3-030-21462-3_8
G. Grätzer, The Congruences of a Finite Lattice. A "Proof-by-Picture" Approach, Birkhäuser, 3rd edition, 2023.
DOI : 10.1007/978-3-031-29063-3_3
I. Rival, Maximal Sublattices of Finite Distributive Lattices, Proc. Amer. Math. Soc. 37 (1973), 417-420.
DOI : 10.1090/s0002-9939-1973-0311527-9
J. Schmid, On maximal sublattices of finite lattices, Discrete Math. 199 (1999), 151-159.
DOI : 10.1016/s0012-365x(98)00207-6
M. Stern, Semimodular lattices: theory and applications, Cambridge University Press, 1999.
DOI : 10.1017/cbo9780511665578

Conjecture on Maximal Sublattices of Finite Semidistributive Lattices and Beyond

K. Adaricheva⁽¹⁾, A. Mata⁽²⁾, S. Silberger⁽¹⁾, A. Zamojska-Dzienio⁽²⁾

(1) Hofstra University

(2) Warsaw University of Technology [WUT]

Download article

Open on arXiv

Submitted on

August 20, 2025

Accepted on

May 19, 2026

Published on

July 23, 2026

Last modified on

July 23, 2026

Volume 28

vol. 28:2

Combinatorics

DOI

10.46298/dmtcs.16374

License

Attribution 4.0 International (CC BY 4.0)

Indicators

163

Views

Downloads

Conjecture on Maximal Sublattices of Finite Semidistributive Lattices and Beyond

K. Adaricheva⁽¹⁾, A. Mata⁽²⁾, S. Silberger⁽¹⁾, A. Zamojska-Dzienio⁽²⁾

(1) Hofstra University

(2) Warsaw University of Technology [WUT]

Abstract

Keywords

Rings and Algebras
06A07, 06A15, 52A37, 52C05, 52C07

Mathematics Subject Classification 2020

06A07 - Combinatorics of partially ordered sets
06A15 - Galois correspondences, closure operators (in relation to ordered sets)
52A37 - Other problems of combinatorial convexity
52C05 - Lattices and convex bodies in \(2\) dimensions (aspects of discrete geometry)
52C07 - Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)

Preview

Loading PDF preview...

References

M. E. Adams, The Frattini sublattice of a distributive lattice, Algebra Universalis 3 (1973), 216-228.
DOI : 10.1007/bf02945121
M. E. Adams, R. Freese, J. B. Nation, J. Schmid, Maximal Sublattices and Frattini Sublattices of Bounded Lattices, J. Austral. Math. Soc., A 63 (1997), 110-127.
DOI : 10.1017/s1446788700000355
K. Adaricheva, J.B. Nation, Convex geometries, in Lattice Theory: Special Topics and Applications, v.2, G. Grätzer and F. Wehrung, eds. Springer, Basel, 2016.
DOI : 10.1007/978-3-319-44236-5_5
K. Adaricheva et al., Convex geometries representable by at most five circles on the plane, In-volve, a Journal of Mathematics 17 (2024), 337-354. Appendices (433 pages) to be found at https://arxiv.org/abs/2008.13077.
DOI : 10.2140/involve.2024.17.337
G. Czédli, Á. Kurusa, A convex combinatorial property of compact sets in the plane and its roots in lattice theory, Categ. Gen. Algebr. Struct. Appl. 11 (2019), 57-92.
DOI : 10.29252/cgasa.11.1.57
C.C. Chen, K.M. Koh, S.K. Tan, Frattini sublattices of distributive lattices, Algebra Universalis 3 (1973), 294-303.
DOI : 10.1007/bf02945131
C.C. Chen, K.M. Koh, S.K. Tan, On the Frattini sublattice of a finite distributive lattice, Algebra Universalis 5 (1975), 88-97.
DOI : 10.1007/bf02485236
A. Day, Doubling construction in lattice theory, Canad. J. Math. 44 (1992), 252-269.
DOI : 10.4153/cjm-1992-017-7
P. H. Edelman, R. E. Jamison, The theory of convex geometries, Geom. Dedicata 19 (1985), 247-270.
DOI : 10.1007/bf00149365
R. Freese, J. Ježek, J.B. Nation, Free Lattices, AMS, 1995.
DOI : 10.1007/bf00403408
B. Ganter, "Properties of finite lattices" by S. Reeg and W. Weiss, revisited, ICFCA 2019, LNAI 11511, D. Cristea et al. (Eds), 99-109.
DOI : 10.1007/978-3-030-21462-3_8
G. Grätzer, The Congruences of a Finite Lattice. A "Proof-by-Picture" Approach, Birkhäuser, 3rd edition, 2023.
DOI : 10.1007/978-3-031-29063-3_3
I. Rival, Maximal Sublattices of Finite Distributive Lattices, Proc. Amer. Math. Soc. 37 (1973), 417-420.
DOI : 10.1090/s0002-9939-1973-0311527-9
J. Schmid, On maximal sublattices of finite lattices, Discrete Math. 199 (1999), 151-159.
DOI : 10.1016/s0012-365x(98)00207-6
M. Stern, Semimodular lattices: theory and applications, Cambridge University Press, 1999.
DOI : 10.1017/cbo9780511665578