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Lattices of homomorphisms

  • B. A. Davey (a1) and H. A. Priestley (a2)

Abstract

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.

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References

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Balbes, R. (1980), ‘Catalytic distributive Lattices’, Algebra Universalis 11, 334340.
Balbes, R. and Dwinger, Ph. (1974), Distributive lattices (University of Missouri Press, Columbia, Missouri).
Cornish, W. H. and Fowler, P. R. (1979), ‘Coproducts of Kleene alglebras’, J. Austral. Math. Soc. (Series A) 27, 209220.
Davey, B. A. (1978), ‘Toplogical duality for prevarieties of universal algebras’, Advances in Math., Suppl. Studies 1, 6199.
Davey, B. A. (1982), ‘Dualities for Stone algebras, double Stone algebras, and relative Stone algebras’, Colloq. Math. 46, 114.
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. 4395).
Davey, B. A. and Priestley, H. A. (1984), ‘Generalized piggyback dualities and applications to Ockham algebras, Houston J. Math., to appear.
Davey, B. A. and Werner, H. (1983a), ‘Dualities and equivalences for varieties of algebras’, Colloq. Math. Soc. János Bolyai 33, 101275.
Davey, B. A. and Werner, H. (1983b), ‘Piggyback dualities’, Colloq. Math. Soc. János Bolyai, to appear.
Davey, B. A. and Werner, H. (1985), ‘Piggyback-Dualitäten’, Bull. Austral. Math. Soc. 32, 132.
Freyd, P. (1966), ‘Algebra valued functors in general and tensor products in particular’, Colloq. Math. 14, 89106.
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).
Goldberg, M. S. (1981), ‘Distributive Ockham algebras: free algebras and injectivity’, Bull. Austral. Math. Soc. 24, 161203.
Kucera, T. G. and Sands, B. (1978), ‘Lattices of lattice homomorphims’, Algebra Universalis 8, 180190.
Priestley, H. A. (1982a), ‘Catalytic distributive lattices and compact zero-dimensional topological lattices’, Algebra Universalis, to appear.
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.
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.
Urquhart, A. (1979), ‘Lattices with a dual homomorphic operation’, Studia Logica 38, 201209.
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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
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