Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-26T11:08:42.376Z Has data issue: false hasContentIssue false
  • English
  • Français

The quest for strong dualities

Part of: General theory of categories and functors Other classes of algebra

Published online by Cambridge University Press:  09 April 2009

David M. Clark  and
Brian A. Davey
David M. Clark
Affiliation:
Mathematics and Computer Science, SUNY, College at New Paltz, NY 12561, USA
Brian A. Davey
Affiliation:
Mathematics, La Trobe University, Bundoora, Victorial 3083, Australia
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We give a revised and updated exposition of the theory of full dualities initiated by Clark, Davey, Krauss and Werner, introducing the (stronger) notion of a strong duality. All known full dualities turn out to be strong. A series of theorems which provide necessary and sufficient conditions for a strong duality to exist is proved. All full dualities in the literature can be obtained from these results and many new strong dualities can be derived. In particular, we show that within congruence distributive varieties every duality can be upgraded to a strong duality. Amongst the new strong dualities are the dualities of Davey, Priestley and Werner for the varieties of pseudocomplemented distributive lattices which are either strong as they stand or can easily be made strong by the addition of partial operations to the dual structures.

Keywords

duality theoryfull dualitystrong dualitycongruence distributivitynear unanimity.

MSC classification

Secondary: 08C05: Categories of algebras 08C15: Quasivarieties 18A40: Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 58 , Issue 2 , April 1995 , pp. 248 - 280
Copyright
Copyright © Australian Mathematical Society 1995

References

[1]Adams, M. E. and Clark, D. M., ‘Endomorphism monoids in minimal quasi primal varieties’, Acta. Sci. Math. (Szeged) 54 (1990), 37–52.Google Scholar
[2]Birkhoff, G., ‘On the combination of subalgebras’, Math. Proc. Cambridge Phil. Soc. 29 (1933), 441–464.CrossRefGoogle Scholar
[3]Clark, D. M., ‘Algebraically and existentially closed Stone and double Stone algebras’, Symbolic Logic 54 (1989), 363–375.CrossRefGoogle Scholar
[4]Clark, D. M. and Davey, B. A., ‘Natural dualities for the working algebraist’, (Cambridge University Press), in preparation.Google Scholar
[5]Clark, D. M. and Krauss, P. H., ‘Topological quasi varieties’, Acta. Sci. Math. (Szeged) 47 (1984), 3–39.Google Scholar
[6]Clark, D. M. and Schmid, J., ‘The countable homogeneous universal model of B 2’, Studia Logica, to appear.Google Scholar
[7]Cornish, W. H. and Fowler, P. R., ‘Coproducts of de Morgan algebras’, Bull. Austral. Math. Soc. 16 (1977), 1–13.CrossRefGoogle Scholar
[8]Davey, B. A., ‘Dualities for equational classes of Brouwerian algebras and Heyting algebras’, Trans. Amer. Math. Soc. 221 (1976), 119–146.CrossRefGoogle Scholar
[9]Davey, B. A., ‘Topological dualities for prevarieties of universal algebras’, in: Studies in foundations and combinatorics (ed. Rota, G.-C.), Adv. Math. Supplementary Studies 1 (Academic Press, Duluth, 1978) pp. 61–99.Google Scholar
[10]Davey, B. A. ‘Dualities for Stone algebras, double Stone algebras, and relative Stone algebras’, Colloq. Math. 46 (1982), 1–14.CrossRefGoogle Scholar
[11]Davey, B. A. ‘Duality theory on ten dollars a day’, in: Algebras and orders (eds. Rosenberg, I. G. and Sabidussi, G.), NATO Advanced Study Institute Series, Series C 389 (Kluwer, Dordrecht, 1993) pp. 71–111.CrossRefGoogle Scholar
[12]Davey, B. A., Heindorf, L. and McKenzie, R., ‘Near unanimity: an obstacle to general duality theory’, Algebra Universalis, to appear.Google Scholar
[13]Davey, B. A. and Priestley, H. A. ‘Optimal natural dualities’, Trans. Amer. Math. Soc. 338 (1993), 655–677.CrossRefGoogle Scholar
[14]Davey, B. A. and Priestley, H. A., ‘Optimal dualities for varieties of Heyting algebras’, Studia Logica, to appear.Google Scholar
[15]Davey, B. A., Quackenbush, R. W. and Schweigert, D., ‘Monotone clones and the varieties they determine’, Order 7 (1990), 145–168.CrossRefGoogle Scholar
[16]Davey, B. A. and Werner, H., ‘Dualities and equivalences for varieties of algebras’, in: Contributions to lattice theory (Szeged, 1980) (eds. Huhn, A. P. and Schmidt, E. T.) Colloq. Math. Soc. J´nos Bolyai 33 (North-Holland, Amsterdam, 1983) pp. 101–275.Google Scholar
[17]Davey, B. A. and Werner, H., ‘Piggyback dualities’, in: Lectures in universal algebra (Szeged, 1983) (eds. Szabó, L. and Szendrei, Á.) Coll. Math. Soc. János Bolyai 43 (North-Holland, Amsterdam, 1986) pp. 61–83.CrossRefGoogle Scholar
[18]Davey, B. A. and Werner, H., ‘Piggyback-Dualitäten’, Bull. Austral. Math. Soc. 32 (1985), 1–32.CrossRefGoogle Scholar
[19]Hofmann, K. H., Mislove, M. and Stralka, A., The Pontryagin duality of compact 0-dimensional semilattices and its applications, Lecture Notes in Mathematics Vol.396 (Springer, Berlin, 1974).CrossRefGoogle Scholar
[20]Jónsson, B., ‘Algebras whose congruence lattices are distributive’, Math. Scand. 21 (1967), 110–121.CrossRefGoogle Scholar
[21]Mitschke, A., ‘Near unanimity identities and congruence distributivity in equational classes’, Algebra Universalis 8 (1978), 29–32.CrossRefGoogle Scholar
[22]Priestley, H. A., ‘Ordered sets and duality for distributive lattices’, Ann. Discrete Math. 23 (1984), 39–60.Google Scholar
[23]Stone, M. H., ‘The theory of representations for Boolean algebras’, Trans. Amer. Math. Soc. 40 (1936), 37–111.Google Scholar