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Locally finite affine complete varieties

Part of: Algebraic structures Varieties

Published online by Cambridge University Press:  09 April 2009

Kalle Kaarli
Kalle Kaarli
Affiliation:
Department of Mathematics University of TartuEE2400 TartuEstonia e-mail: Kaarli@math.ut.ee
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Abstract

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The main results of the paper are the following: 1. Every locally finite affine complete variety admits a near unanimity term; 2. A locally finite congruence distributive variety is affine complete if and only if all its algebras with no proper subalgebras are affine complete and the variety is generated by one of such algebras. The first of these results sharpens a result of McKenzie asserting that all locally finite affine complete varieties are congruence distributive. The second one generalizes the result by Kaarli and Pixley that characterizes arithmetical affine complete varieties.

Keywords

Affine complete varietiescongruence distributivitynear unanimity functionscompatible function systems

MSC classification

Secondary: 08B10: Congruence modularity, congruence distributivity 08A40: Operations, polynomials, primal algebras

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 62 , Issue 2 , April 1997 , pp. 141 - 159
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Baker, K. and Pixley, A., ‘Polynomial interpolation and the Chinese remainder theorem for algebraic systems’, Math. Z. 143 (1975), 165–174.Google Scholar
[2]Davey, B. A. and Werner, H., ‘Dualities and equivalences of varieties of algebras’, in: Contributions to lattice theory (Proc. Conf. Szeged. 1980) Colloq. Math. Soc. János Bolyai 33 (North-Holland, Amsterdam, 1983) pp. 101–275.Google Scholar
[3]Grätzer, G., ‘On Boolean functions. (Notes on lattice theory. II.)’, Revue de Math. Pure et Appliquées 7 (1962), 693–697.Google Scholar
[4]Hobby, D. and McKenzie, R., The structure of finite algebras, Contemp. Math. 76 (Amer. Math. Soc., Providence, 1988).CrossRefGoogle Scholar
[5]Hu, T. -K., ‘Characterization of algebraic functions in equational classes generated by independent primal algebras’, Algebra Universalis 17 (1983), 200–207.Google Scholar
[6]Kaarli, K., ‘On varieties generated by functionally complete algebras’, Algebra Universalis 29 (1992), 495–502.Google Scholar
[7]Kaarli, K., ‘On affine complete varieties generated by hemiprimal algebras with Boolean congruence lattices’, Tartu Riikl. Ül. Toimetised 878 (1990), 23–38.Google Scholar
[8]Kaarli, K. and Pixley, A., ‘Affine complete varieties’, Algebra Universalis 20 (1987), 74–90.Google Scholar
[9]Kaarli, K., and Pixley, A. ‘Finite arithemetical affine complete algebras having no proper subalgebras’, Algebra Universalis 31 (1994), 557–579.Google Scholar
[10]Mitschke, A., ‘Near unanimity identities and congruence distributivity of equational classes’, Algebra Universalis 8 (1978), 29–32.Google Scholar
[11]Pixley, A., ‘Completeness in arithmetical varieties’, Algebra Universalis 2 (1972), 179–196.Google Scholar
[12]Pixley, A., ‘Functional and affine completeness and arithmetical varieties‘, in: Proc. NATO Advanced Studies Institute, Series C, Vol. 389 (Kluwer, Dordrecht) pp. 317–357.Google Scholar