Abstract
It was shown in [3] (see also [5]) that there is a duality between the category of bounded distributive lattices endowed with a join-homomorphism and the category of Priestley spaces endowed with a Priestley relation. In this paper, bounded distributive lattices endowed with a join-homomorphism, are considered as algebras and we characterize the congruences of these algebras in terms of the mentioned duality and certain closed subsets of Priestley spaces. This enable us to characterize the simple and subdirectly irreducible algebras. In particular, Priestley relations enable us to characterize the congruence lattice of the Q-distributive lattices considered in [4]. Moreover, these results give us an effective method to characterize the simple and subdirectly irreducible monadic De Morgan algebras [7].
The duality considered in [4], was obtained in terms of the range of the quantifiers, and such a duality was enough to obtain the simple and subdirectly irreducible algebras, but not to characterize the congruences.
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. and Ph. Dwinger, 1975, ‘Equational classes of closure algebras’, Ind. Math. 37, 189–198.
Cignoli, R., 1991, ‘Distributive lattice congruences and Priestley spaces’, Actas del primer congreso Dr. Antonio Monteiro, Univeisidad Nacional del Sur, Bahía Blanca, 81–84.
Cignoli, R., S. Lafalce and A. Petrovich, 1991, ‘Remarks on Priestley duality for distributive lattices’, Order 8, 299–315.
Cignoli, R., 1991, ‘Quantifiers on distributive lattices’, Discrete Math. 96, 183–197.
Goldblatt, R., 1989, ‘Varieties of Complex algebras’, Ann. Pure Appl. Logic. 44, 173–242.
McKinsey, J. C. C. and A. Tarski, 1944, ‘The algebra of topology’, Ann. of Math. 45, 141–191.
Petrovich, A., ‘Monadic De Morgan algebras’ (to appear).
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. 3, 507–530.
Priestley, H. A., 1974, ‘Stone lattices: a topological approach’, Fund. Math. 84, 127–143.
Priestley, H. A., 1984, ‘Ordered sets and duality for distributive lattices’, Ann. Discrete Math. 23, 39–60.
Rasiowa, H. and R. Sikorski, 1963, The mathematics of metamathematics, Warszawa.
Author information
Authors and Affiliations
Additional information
I would like to thank my research supervisor Dr. Roberto Cignoli for his helpful suggestions during the preparation of this paper and the referee for calling my attention to Goldblatt's paper [5].
Rights and permissions
About this article
Cite this article
Petrovich, A. Distributive lattices with an operator. Stud Logica 56, 205–224 (1996). https://doi.org/10.1007/BF00370147
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00370147