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EVERY SHIFT AUTOMORPHISM VARIETY HAS AN INFINITE SUBDIRECTLY IRREDUCIBLE MEMBER

Part of: Model theory Varieties

Published online by Cambridge University Press:  19 January 2010

KATE S. OWENS
KATE S. OWENS*
Affiliation:
Department of Mathematics, University of South Carolina, 1523 Greene Street, Columbia, SC 29208, USA (email: [email protected])
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Abstract

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A shift automorphism algebra is one satisfying the conditions of the shift automorphism theorem, and a shift automorphism variety is a variety generated by a shift automorphism algebra. In this paper, we show that every shift automorphism variety contains a countably infinite subdirectly irreducible algebra.

Keywords

shift automorphism algebrashift automorphism varietyinfinite subdirectly irreducible algebras

MSC classification

Secondary: 03C05: Equational classes, universal algebra 08B26: Subdirect products and subdirect irreducibility
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 88 , Issue 2 , April 2010 , pp. 231 - 238
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

References

[1]Bajusz, T., McNulty, G. F. and Szendrei, Á., ‘Lyndon’s groupoid is not inherently nonfinitely based’, Algebra Universalis 27(2) (1990), 254–260.CrossRefGoogle Scholar
[2]Baker, K. A., ‘Finite equational bases for finite algebras in a congruence-distributive equational class’, Adv. Math. 24(3) (1977), 207–243.CrossRefGoogle Scholar
[3]Baker, K. A., McNulty, G. F. and Werner, H., ‘The finitely based varieties of graph algebras’, Acta Sci. Math. (Szeged) 51(1–2) (1987), 3–15.Google Scholar
[4]Baker, K. A., McNulty, G. F. and Werner, H., ‘Shift-automorphism methods for inherently nonfinitely based varieties of algebras’, Czechoslovak Math. J. 39(114)(1) (1989), 53–69.CrossRefGoogle Scholar
[5]Berman, J., ‘A proof of Lyndon’s finite basis theorem’, Discrete Math. 29(3) (1980), 229–233.CrossRefGoogle Scholar
[6]Dziobiak, W., ‘On infinite subdirectly irreducible algebras in locally finite equational classes’, Algebra Universalis 13(3) (1981), 393–394.CrossRefGoogle Scholar
[7]Kearnes, K. and Willard, R., ‘Inherently nonfinitely based solvable algebras’, Canad. Math. Bull. 37(4) (1994), 514–521.CrossRefGoogle Scholar
[8]Lyndon, R. C., ‘Identities in two-valued calculi’, Trans. Amer. Math. Soc. 71 (1951), 457–465.CrossRefGoogle Scholar
[9]Lyndon, R. C., ‘Identities in finite algebras’, Proc. Amer. Math. Soc. 5 (1954), 8–9.CrossRefGoogle Scholar
[10]McKenzie, R., ‘Para primal varieties: a study of finite axiomatizability and definable principal congruences in locally finite varieties’, Algebra Universalis 8(3) (1978), 336–348.CrossRefGoogle Scholar
[11]McKenzie, R., ‘The residual bounds of finite algebras’, Internat. J. Algebra Comput. 6 (1996), 1–28.CrossRefGoogle Scholar
[12]McNulty, G. F., Székely, Z. and Willard, R., ‘Equational complexity of the finite algebra membership problem’, Internat. J. Algebra Comput. 18 (2008), 1283–1319.CrossRefGoogle Scholar
[13]Murskiĭ, V. L., ‘The existence in three-valued logic of closed class with finite basis not having a finite complete set of identities’, English Translation Soviet Math. Dokl. 6 (1965), 1020–1024.Google Scholar
[14]Murskiĭ, V. L., ‘The number of k-element algebras with a binary operation which do not have a finite basis of identities’, Problemy Kibernet. (1979), 5–27.Google Scholar
[15]Park, R. E., ‘Equational classes on non-associative ordered algebras’, PhD Thesis, University of California, Los Angeles, 1976.Google Scholar
[16]Post, E. L., The Two-Valued Iterative Systems of Mathematical Logic, Annals of Mathematics Studies, 5 (Princeton University Press, Princeton, NJ, 1941).Google Scholar
[17]Quackenbush, R. W., ‘Equational classes generated by finite algebras’, Algebra Universalis 1 (1971/72), 265–266.CrossRefGoogle Scholar
[18]Szendrei, Á., ‘Non-finitely based finite groupoids generating minimal varieties’, Acta Sci. Math. (Szeged) 57(1–4) (1993), 593–600.Google Scholar
[19]Wald, L. A., ‘Minimal inherently nonfinitely based varieties of groupoids’ PhD Thesis, University of California, Los Angeles, 1998.Google Scholar
[20]Walter, B. L., ‘The finitely based varieties of looped directed graph algebras’, Acta Sci. Math. (Szeged) 72(3–4) (2006), 421–458.Google Scholar
[21]Willard, R., ‘The finite basis problem’, in: Contributions to General Algebra, Vol. 15 (Heyn, Klagenfurt, 2004), pp. 199–206.Google Scholar