Abstract
A modal lattice is a bounded distributive lattice endowed with a unary operator which preserves the join-operation and the smallest element. In this paper we consider the variety CH of modal lattices that is generated by the totally ordered modal lattices and we characterize the lattice of subvarieties of CH. We also give an equational basis for each subvariety of CH.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Blok, W. J. (1980) The lattice of modal logics: an algebraic investigation, J.S.L. 45(2), 221–236.
Blok, W. J. (1980) The lattice of varieties of modal algebras is not strongly atomic, Algebra Universalis 11, 285–294.
Blok, W. J. (1980) Pretabular varieties of modal algebras, Studia Logica 39, 101–124.
Makinson, D. C. (1971) Some embedding theorems for modal logic, Notre Dame J. Formal Logic 12, 252–254.
Cignoli, R., Lafalce, S. and Petrovich, A. (1991) Remarks on Priestley duality for distributive lattices, Order 8, 299–315.
Goldblatt, R. (1989) Varieties of complex algebras, Ann. Pure Appl. Logic 44(3), 153–301.
Makinson, D. (1971) Aspectos de la lógica modal, Notas de Lógica Matemática, Instituto de Matemática, Universidad Nacional del Sur, Bahía Blanca, 28.
Petrovich, A. (1996) Distributive lattices with an operator, Studia Logica 56, 205–224.
Priestley, H. A. (1970) Representation of distributive lattices by means of ordered Stone spaces, Bull. London Math. Soc. 2, 186–190.
Priestley, H. A. (1972) Ordered topological spaces and the representation of distributive lattices, Proc. London Math. Soc. 2(4), 507–530.
Priestley, H. A. (1974) Stone lattices: A topological approach, Fund. Math. 84, 127–143.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Petrovich, A. Equational Classes of Totally Ordered Modal Lattices. Order 16, 1–17 (1999). https://doi.org/10.1023/A:1006259631226
Issue Date:
DOI: https://doi.org/10.1023/A:1006259631226