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Lattices of homomorphisms

Part of: Categories with structure Distributive lattices Algebraic logic Boolean algebras (Boolean rings) Special properties

Published online by Cambridge University Press:  09 April 2009

B. A. Davey  and
H. A. Priestley
B. A. Davey
Affiliation:
Department of Pure Mathematics, La Trobe University, Bundoora, Victoria 3083, Australia
H. A. Priestley
Affiliation:
Mathematical Institute, 24/29 St. Giles, Oxford OX1 3LB, England
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Abstract

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Given a variety K of lattice-ordered algebras, A ∈ K is catalytic if for all B ∈ K, K(A, B) is a lattice for the pointwise order. The catalytic objects are determined for various varieties of distributive-lattice-ordered algebras. The characterisations obtained do not show an overall unity and exhibit diverse behaviour. Duality is employed extensively. Its usefulness in this context depends on the existence of an order-isomorphism between K(A, B) and the corresponding dual horn-set. Criteria for the existence of such an order-isomorphism are investigated for dualities of the Davey-Werner type. The relationship between catalytic objects and colattices is also discussed.

MSC classification

Secondary: 03G10: Lattices and related structures 06D15: Pseudocomplemented lattices 18D35: Structured objects in a category (group objects, etc.) 54F05: Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 40 , Issue 3 , June 1986 , pp. 364 - 406
Copyright
Copyright © Australian Mathematical Society 1986

References

Balbes, R. (1980), ‘Catalytic distributive Lattices’, Algebra Universalis 11, 334–340.CrossRefGoogle Scholar
Balbes, R. and Dwinger, Ph. (1974), Distributive lattices (University of Missouri Press, Columbia, Missouri).Google Scholar
Cornish, W. H. and Fowler, P. R. (1979), ‘Coproducts of Kleene alglebras’, J. Austral. Math. Soc. (Series A) 27, 209–220.CrossRefGoogle Scholar
Davey, B. A. (1978), ‘Toplogical duality for prevarieties of universal algebras’, Advances in Math., Suppl. Studies 1, 61–99.Google Scholar
Davey, B. A. (1982), ‘Dualities for Stone algebras, double Stone algebras, and relative Stone algebras’, Colloq. Math. 46, 1–14.CrossRefGoogle Scholar
Davey, B. A. and Duffus, D. (1982), ‘Exponentiation and duality’ (in Ordered sets, edited by Rival, I., NATO Advanced Study Institutes Series, D. Riedel Publishing Company, Dordrecht, pp. 43–95).CrossRefGoogle Scholar
Davey, B. A. and Priestley, H. A. (1984), ‘Generalized piggyback dualities and applications to Ockham algebras, Houston J. Math., to appear.Google Scholar
Davey, B. A. and Werner, H. (1983a), ‘Dualities and equivalences for varieties of algebras’, Colloq. Math. Soc. János Bolyai 33, 101–275.Google Scholar
Davey, B. A. and Werner, H. (1983b), ‘Piggyback dualities’, Colloq. Math. Soc. János Bolyai, to appear.Google Scholar
Davey, B. A. and Werner, H. (1985), ‘Piggyback-Dualitäten’, Bull. Austral. Math. Soc. 32, 1–32.CrossRefGoogle Scholar
Freyd, P. (1966), ‘Algebra valued functors in general and tensor products in particular’, Colloq. Math. 14, 89–106.CrossRefGoogle Scholar
Gierz, G., Hofmann, K. H., Keimel, K., Lawson, J. D., Mislove, M. and Scott, D. S. (1980), A compendium of continuous lattices, (Springer-Verlag, Berlin, Heidelberg, New York).CrossRefGoogle Scholar
Goldberg, M. S. (1981), ‘Distributive Ockham algebras: free algebras and injectivity’, Bull. Austral. Math. Soc. 24, 161–203.CrossRefGoogle Scholar
Kucera, T. G. and Sands, B. (1978), ‘Lattices of lattice homomorphims’, Algebra Universalis 8, 180–190.CrossRefGoogle Scholar
Priestley, H. A. (1982a), ‘Catalytic distributive lattices and compact zero-dimensional topological lattices’, Algebra Universalis, to appear.Google Scholar
Priestley, H. A. (1982b), ‘Algebraic lattices as duals of distributive lattices’ (in Proceedings of the conference on topological and categorical aspects of continuous lattices, Bremen, 1982, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York), to appear.Google Scholar
Priestley, H. A. (1982c), ‘Ordered sets and duality for distributive lattices’ (in Proceedings of the conference on ordered sets and their applications, Lyon, 1982, Annals of Discrete Mathematics, North-Holland, Amsterdam, London), to appear.Google Scholar
Urquhart, A. (1979), ‘Lattices with a dual homomorphic operation’, Studia Logica 38, 201–209.CrossRefGoogle Scholar