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Minimal partial clones

Published online by Cambridge University Press:  17 April 2009

F. Börner ,
L. Haddad  and
R. Pöschel
F. Börner
Affiliation:
Karl-Weierstrass-Institut for Mathematics, Mohrenstrasse 39, Berlin 1086, Germany
L. Haddad
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario, CanadaM5S 1A1
R. Pöschel
Affiliation:
Karl-Weierstrass-Institut for Mathematics, Mohrenstrasse 39, Berlin 1086, Germany
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Abstract

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Let A be a finite set. A partial clone on A is a composition closed set of operations containing all projections. It is well known that the partial clones on A, ordered by inclusion, form a lattice. We show that the minimal partial clones on A are:

(a) minimal clones of full operations or

(b) generated by partial projections defined on a totally reflexive and totally symmetric domain.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 44 , Issue 3 , December 1991 , pp. 405 - 415
Copyright
Copyright © Australian Mathematical Society 1991

References

[1]Börner, F., Operationen auf Relationen, Dissertation (Universität Leipzig, 1988).Google Scholar
[2]Burmeister, P., A model theoretic oriented approach to partial algebras (Introduction to theory and applications of partial algebras Part I) 32 (Math. Research, Akademie Verlag Berlin, 1986).Google Scholar
[3]Csákány, B., ‘All minimal clones on the three-element set’, Acta Cybernet. 6 (1983), 227–238.Google Scholar
[4]Haddad, L. and Rosenberg, I.G., ‘Completeness theory for finite partial algebras’, Algebra Universalis (to appear).Google Scholar
[5]Haddad, L., Rosenberg, I.G. and Schweigert, D., ‘A maximal partial clone and a Slupecki type criterion’, Acta. Sci. Math. 54 (1990), 89–98.Google Scholar
[6]Pálfy, P.P., ‘Minimal clones’, preprint 27/1984, Math. Inst. Hungarian Acad. Sci. (1984).Google Scholar
[7]Pálfy, P.P., The arity of minimal clones, Acta Sci. Math. 50 (1986), 331–333.Google Scholar
[8]Pöschel, R. and Kalužnin, L.A., Funktionen und Relationenalgebren. Ein Kapitel der diskreten Mathematik (Deutscher Verlag der Wiss., Berlin, 1979 and Birkhauser, Basel, Stuttgart 1979).CrossRefGoogle Scholar
[9]Post, E.L., ‘The two-valued iterative systems of mathematical logic’, Ann. Math. Studies 5 (1941).Google Scholar
[10]Rosenberg, I.G., ‘La structure des fonctions de plusieurs variables sur un ensemble fini’, C.R. Acad. Sci. Paris Sér. A-B 260 (1965), 3817–3819.Google Scholar
[11]Rosenberg, I.G., ‘Über die funktionale Vollständigkeit in dem mehrwertigen Logiken’, Rozpravy Čs. Akademie Věd. Ser. Math. Nat. Sci. 80 (1970), 3–93.Google Scholar
[12]Rosenberg, I.G., ‘Minimal clones I: The five types’, in Lectures in Universal Algebra 43, Colloq. Math. Soc. J. Bolyai, pp. 405–427, 1983.CrossRefGoogle Scholar
[13]Szendrei, A., Clones in universal algebra 99, Séminaire de Mathématiques Supérieures (Université de Montréal, Canada, 1986).Google Scholar