Modal Logic of Lattices

Maciej A. Hałapacz

doi:10.18778/0138-0680.2026.10

Authors

Maciej A. Hałapacz University of Wrocław, Department of Logic and Methodology of Sciences, Wrocław, Poland Author https://orcid.org/0000-0001-8638-8488

DOI:

https://doi.org/10.18778/0138-0680.2026.10

Keywords:

modal logics of classes of structures, lattices, order, distributivity

Abstract

We prove that the modal logic of lattices with the accessibility relation of being isomorphic to a sublattice is S4.2. The same is proven for modular and distributive lattices.

References

S. Berger, A. C. Block, B. Löwe, The modal logic of abelian groups, Algebra Universalis, vol. 84 (2023), p. 32, DOI: https://doi.org/10.1007/s00012-023-00821-9.

P. Blackburn, M. de Rijke, Y. Venema, Modal Logic, vol. 53 of Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, Cambridge (2001), DOI: https://doi.org/10.1017/CBO9781107050884.

J. Grygiel, The Concept of Gluing for Lattices, Jan Długosz University of Częstochowa, Częstochowa (2004).

J. D. Hamkins, G. Leibman, B. Löwe, Structural connections between a forcing class and its modal logic, Israel Journal of Mathematics, vol. 207(2) (2015), pp. 617–651, DOI: https://doi.org/10.1007/s11856-015-1185-5.

J. D. Hamkins, B. Löwe, The modal logic of forcing, Transactions of the American Mathematical Society, vol. 360(4) (2008), pp. 1793–1817, DOI: https://doi.org/10.1090/S0002-9947-07-04297-3.

J. D. Hamkins, W. A. Wołoszyn, The modal model theory, Notre Dame Journal of Formal Logic, vol. 65(1) (2024), pp. 1–37, DOI: https://doi.org/10.1215/00294527-2024-0001.

T. Inamdar, B. Löwe, The modal logic of inner models, Journal of Symbolic Logic, vol. 82(1) (2017), pp. 225–236, DOI: https://doi.org/10.1017/jsl.2016.6.

J. Kotas, P. Wojtylak, Finite distributive lattices as sums of Boolean algebras, Reports on Mathematical Logic, vol. 29 (1995), pp. 35–40.

D. I. Saveliev, I. Shapirovsky, On modal logics of model-theoretic relations,Studia Logica, vol. 108(5) (2020), pp. 989–1017, DOI: https://doi.org/10.1007/s11225-019-09885-y.

A. Wroński, Remarks on intermediate logics with axioms containing only one variable, Bulletin of the Section of Logic, vol. 2(1) (1973), pp. 58–64.