Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-16T18:23:18.324Z Has data issue: false hasContentIssue false
  • English
  • Français

Natural duality via a finite set of relations

Published online by Cambridge University Press:  17 April 2009

László Zádori
László Zádori
Affiliation:
JATE, Bolyai Intézet, Aradi Vértanúk Tere 1, H-6720 Szeged, Hungary
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 present a duality theorem. We give a necessary and sufficient condition for any set of algebraic relations to entail the set of all algebraic relations in Davey and Werner's sense. The main result of the paper states that for a finite algebra a finite set of algebraic relations yields a duality if and only if the set of all algebraic relations can be obtained from it by using four types of relational constructs. Finally, we prove that a finite algebra admits a natural duality if and only if the algebra has a near unanimity term operation, provided that the algebra possesses certain 2k-ary term operations for some k. This is a generalisation of a theorem of Davey, Heindorf and McKenzie.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 51 , Issue 3 , June 1995 , pp. 469 - 478
Copyright
Copyright © Australian Mathematical Society 1995

References

[1]Bodnarĉuk, V.G., Kalužnin, L.A., Kotov, V.A. and Romov, V.A., ‘Galois theory for Post algebras I, II’, (in Russian), Kibernetika (Kiev) 5 (1969), no. 3, 1–10, no. 5, 1–9.Google Scholar
[2]Clark, D.M. and Krauss, P.H., ‘Topological quasi varieties’, Acta Sci. Math. 47 (1984), 3–39.Google Scholar
[3]Clark, D.M. and Davey, B.A., ‘The quest for strong dualities’, J. Austral. Math. Soc. (to appear).Google Scholar
[4]Davey, B.A. and Werner, H., ‘Dualities and equivalences for varieties of algebras’, Colloq. Math. Soc. János Bolyai 33 (1983), 101–275.Google Scholar
[5]Davey, B.A. and Priestley, H.A., ‘Generalized piggyback dualities and applications to Ockham algebras’, Houston J. Math. 13 (1987), 151–197.Google Scholar
[6]Davey, B.A., ‘Duality theory on ten dollars a Day’, in Algebras and orders, (Rosenberg, I.G. and Sabidussi, G., Editors), NATO Advanced Study Institute Series, Series C 398 (Kluwer Academic Publishers, 1993), pp. 71–111.CrossRefGoogle Scholar
[7]Davey, B.A., Heindorf, L. and McKenzie, R., ‘Near unanimity: an obstacle to general duality theory’, Algebra Universalis (to appear).Google Scholar
[8]Davey, B.A. and Rosenberg, I.G., ‘Algebraic duality’, (preprint).Google Scholar
[9]Davey, B.A., Haviar, M. and Priestley, H.A., ‘The syntax and semantics of entailment in duality theory’, (preprint).Google Scholar
[10]Geiger, D., ‘Closed systems of functions and predicates’, Pacific J. Math. 27 (1968), 95–100.CrossRefGoogle Scholar
[11]Willard, R.D., Personal communication.Google Scholar