Abstract
We study the existence of nontrivial and of representable (dual) weak complementations, along with the lattice congruences that preserve them, in different constructions of bounded lattices, then use this study to determine the finite (dual) weakly complemented lattices with the largest numbers of congruences, along with the structures of their congruence lattices. It turns out that, if \(n\ge 7\) is a natural number, then the four largest numbers of congruences of the n–element (dual) weakly complemented lattices are: \(2^{n-2}+1\), \(2^{n-3}+1\), \(5\cdot 2^{n-6}+1\) and \(2^{n-4}+1\), which yields the fact that, for any \(n\ge 5\), the largest and second largest numbers of congruences of the n–element weakly dicomplemented lattices are \(2^{n-3}+1\) and \(2^{n-4}+1\). For smaller numbers of elements, several intermediate numbers of congruences appear between the elements of these sequences.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bishop, A.A.: Completions of Lattices with Semicomplementation. Dissertation, Western Michigan University (1972)
Czédli, G., Mureşan, C.: On Principal Congruences and the Number of Congruences of a Lattice with More Ideals than Filters. Acta Universitatis Szegediensis, Acta Scientiarum Mathematicarum (2019)
Czédli, G.: A Note on Finite Lattices with Many Congruences, Acta Universitatis Matthiae Belii. Series Mathematics Online , 22–28 (2018)
Czédli, G.: Finite Semilattices with Many Congruences. Order 36(2), 233–247 (2019)
Freese, R.: Computing Congruence Lattices of Finite Lattices. Proc. Amer. Math. Soc. 125, 3457–3463 (1997)
Ganter, B., Wille, R.: Formal Concept Analysis. Springer, Mathematical Foundations (1999)
Ganter, B., Kwuida, L.: Finite Distributive Concept Algebras. Order 23, 235–248 (2006)
Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.S.: Continuous lattices and domains, Encyclopedia of Mathematics and its Applications, vol. 93, pp xxxvi+591. Cambridge University Press, Cambridge (2003)
Giuntini, R., Mureşan, C., Paoli, F., PBZ* –lattices: Ordinal and Horizontal Sums, Algebraic Perspectives on Substructural Logics, Trends in Logic Series, vol. 55, pp. 73–105. Springer Nature (2020)
Gorbunov, V.A.: Algebraic Theory of Quasivarieties (translated from Russian), Siberian School of Algebra and Logic, pp. xii+298. Consultants Bureau, New York (1998)
Grätzer, G.: General Lattice Theory. Birkhäuser Akademie-Verlag, Basel-Boston-Berlin (1978)
Grätzer, G.: Universal Algebra, 2nd edn. Springer Science+Business Media, LLC, New York (2008)
Grätzer, G.: The Congruences of Finite Lattices. A ”proof–by–picture“ Approach, Second ed.. Birkhäuser–Springer, Cham (2016)
Kwuida, L.: Congruences of Concept Algebras, Proceedings of Novi Sad Algebraic Conference 2003. Novi. Sad. J. Math. 34(2), 141–152 (2004)
Kwuida, L.: Dicomplemented Lattices. A Contextual Generalization of Boolean Algebras. Dissertation, TU Dresden, Shaker–Verlag (2004)
Kwuida, L., Machida, H.: On the Isomorphism Problem of Weakly Dicomplemented Lattices. Ann. Math. Artif. Intell. 59(2), 223–239 (2010)
Mureşan, C.: On the Cardinalities of the Sets of Congruences, Ideals and Filters of a Lattice, Analele Universităţii Bucureşti. Seria Informatică. Proceedings of the Workshop Days of Computer Science (DACS): LXII, affiliated workshop of the 11th edition of the conference Computability in Europe (2015), 55–68. University of Bucharest, Bucharest, Romania (2015)
Mureşan, C.: Cancelling Congruences of Lattices, While Kee** Their Numbers of Filters and Ideals, South American Journal of Logic (2020)
Mureşan, C., Kulin, J.: On the Largest Numbers of Congruences of Finite Lattices. Order 37(3), 445–460 (2020)
Mureşan, C.: Some Properties of Lattice Congruences Preserving Involutions and Their Largest Numbers in the Finite Case, Houston Journal of Mathematics 47(2), 295–320 (2021)
Mureşan, C.: A Note on Congruences of Infinite Bounded Involution Lattices. Scientific Annals of Computer Science 31(1), 51–78 (2021)
Szász, G.: On Weakly Complemented Lattices. Acta Sci. Math. (Szeged) 16(1–2), 122–126 (1955)
Trullenque Ortiz, C.: Complete Theories of Boolean Algebras. Grade Thesis, University of Barcelona, Faculty of Mathematics and Computer Science (2018)
Wille, R.: Boolean Concept Logic. In Ganter, B., Mineau, G.W. (eds.) ICCS 2000, Conceptual Structures: Logical, Linguistic, and Computational Issues, vol. 1867, pp. 317–331. Springer LNAI (2000)
Acknowledgements
This work was supported by the research grant number IZSEZO_186586/1, awarded to the project Reticulations of Concept Algebras by the Swiss National Science Foundation, within the programme Scientific Exchanges. We thank the anonymous referee for his or her good suggestions for improving the form of our paper.
Funding
Open access funding provided by Bern University of Applied Sciences
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Kwuida, L., Mureşan, C. On Nontrivial Weak Dicomplementations and the Lattice Congruences that Preserve Them. Order 40, 423–453 (2023). https://doi.org/10.1007/s11083-022-09597-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11083-022-09597-4
Keywords
- (principal) congruence
- (co)atom of a bounded lattice
- (glued
- ordinal
- horizontal) sum of bounded lattices
- (nontrivial
- representable) (dual) weak complementation