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Simple surjective algebras having no proper subalgebras

Part of: Algebraic structures Varieties

Published online by Cambridge University Press:  09 April 2009

Ágnes Szendrei
Ágnes Szendrei
Affiliation:
Bolyai InstituteAradi vértanúk tere 1 6720 Szeged, Hungary
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Abstract

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We prove that every finite, simple, surjective algebra having no proper subalgebras is either quasiprimal or affine or isomorphic to an algebra term equivalent to a matrix power of a unary permutational algebra. Consequently, it generates a minimal variety if and only if it is quasiprimal. We show also that a locally finite, minimal variety omitting type 1 is minimal as a quasivariety if and only if it has a unique subdirectly irreducible algebra.

MSC classification

Secondary: 08A05: Structure theory 08A40: Operations, polynomials, primal algebras 08B05: Equational logic, Mal′cev (Mal′tsev) conditions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 48 , Issue 3 , June 1990 , pp. 434 - 454
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Bergman, C. and McKenzie, R., ‘Minimal varieties and quasivarieties’, J. Austral. Math. Soc., to appear.Google Scholar
[2]Clark, D. M. and Krauss, P. H., ‘Plain para primal algebras’, Algebra Universalis 11 (1980), 365–388.CrossRefGoogle Scholar
[3]Csákány, B., ‘Completeness in coalgebras’, Acta Sci. Math. (Szeged) 48 (1985), 75–84.Google Scholar
[4]Hobby, D. and McKenzie, R., The structure of finite algebras (Tame congruence theory), (Contemp. Math., vol. 76, Amer. Math. Soc., Providence, R. I., 1988).CrossRefGoogle Scholar
[5]Jónsson, B., ‘Algebras whose congruence lattices are distributive’, Math. Scand. 21 (1967), 110–121.CrossRefGoogle Scholar
[6]McKenzie, R., ‘On minimal, locally finite varieties with permuting congruence relations’, preprint, 1976.Google Scholar
[7]McKenzie, R., ‘Finite forbidden lattices’, Universal algebra and lattice theory, Proc. Conf. Puebla, 1982, pp. 176–205 (Lecture Notes in Math. 1004, Springer-Verlag, 1983).Google Scholar
[8]Pálfy, P. P. and Szendrei, Á., ‘Unary polynomials in algebras II’, Contributions to general algebra, Proc. Klagenfurt Conf., 1982, pp. 273–290 (Verlag Hölder-Pichler-Tempsky, Wien, Verlag Teubner, Stuttgart, 1983).Google Scholar
[9]Pixley, A. F., ‘Functionally complete algebras generating distributive and permutable classes’, Math. Z. 114 (1970), 361–372.Google Scholar
[10]Pixley, A. F., ‘The ternary discriminator function in universal algebra’, Math. Ann. 191 (1971), 167–180.CrossRefGoogle Scholar
[11]Salomaa, A. A., ‘A theorem concerning the composition of functions of several variables ranging over a finite set’, J. Symbolic Logic 25 (1960), 203–208.CrossRefGoogle Scholar
[12]Slupecki, J., ‘Completeness criterion for systems of many-valued propositional calculus’, C. R. des Séances de la Société des Sciences et des Lettres de Varsovie Cl. II 32 (1939), 102–109 (Polish); English transl., Studia Logica 30 (1972), 153–157.Google Scholar
[13]Smith, J. D. H., Mal'cev varieties, (Lecture Notes in Math. 554, Springer-Verlag, Berlin, 1976).CrossRefGoogle Scholar
[14]Szendrei, Á., Clones in universal algebra (Séminaire de Mathématiques Supérieures, vol. 99, Les Presses de l' Université de Montréal, Montréal, 1986).Google Scholar
[15]Szendrei, Á., ‘Idempotent algebras with restrictions on subalgebras’, Acta Sci. Math. (Szeged) 51 (1987), 251–268.Google Scholar
[16]Szendrei, Á., ‘The primal algebra characterization theorem revisited’, Algebra Universalis, submitted.Google Scholar
[17]Taylor, W., ‘The fine spectrum of a variety’, Algebra Universalis 5 (1975), 262–303.Google Scholar