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Chief Factor Sizes in Finitely Generated Varieties

Published online by Cambridge University Press:  20 November 2018

K. A. Kearnes ,
E. W. Kiss ,
Á. Szendrei  and
R. D. Willard
K. A. Kearnes
Affiliation:
Department of Mathematics, University of Colorado, Boulder, CO 80309-0395, USA, email: [email protected]
E. W. Kiss
Affiliation:
Eötvös University, Department of Algebra and Number Theory, Múzeum krt. 6–8, 1088 Budapest, Hungary, email: [email protected]
Á. Szendrei
Affiliation:
Bolyai Institute, Aradi vértanúk tere 1, H-6720 Szeged, Hungary, email: [email protected]
R. D. Willard
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, email: [email protected]
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Abstract

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Let $\mathbf{A}$ be a $k$-element algebra whose chief factor size is $c$. We show that if $\mathbf{B}$ is in the variety generated by $\mathbf{A}$, then any abelian chief factor of $\mathbf{B}$ that is not strongly abelian has size at most ${{c}^{k-1}}$. This solves Problem 5 of The Structure of Finite Algebras, by D. Hobby and R. McKenzie. We refine this bound to $c$ in the situation where the variety generated by $\mathbf{A}$ omits type 1. As a generalization, we bound the size of multitraces of types 1, 2, and 3 by extending coordinatization theory. Finally, we exhibit some examples of bad behavior, even in varieties satisfying a congruence identity.

Keywords

08B26tame congruence theorychief factormultitrace
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 54 , Issue 4 , 01 August 2002 , pp. 736 - 756
Copyright
Copyright © Canadian Mathematical Society 2002

References

[1] Burris, S. and Sankappanavar, H. P., A Course in Universal Algebra. Springer-Verlag, 1981.Google Scholar
[2] Freese, R. and McKenzie, R., Residually small varieties with modular congruence lattices. Trans. Amer. Math. Soc. 264 (1981), 419–430.Google Scholar
[3] Freese, R. and McKenzie, R., Commutator Theory for Congruence Modular Varieties. LondonMath. Soc. Lecture Note Ser. 125, Cambridge University Press, Cambridge-New York, 1987.Google Scholar
[4] Hobby, D. and McKenzie, R., The Structure of Finite Algebras. Contemp.Math. 76, Amer.Math. Soc., 1988.Google Scholar
[5] Kearnes, K. A., Kiss, E. W., and Valeriote, M. A., Minimal sets and varieties. Trans. Amer.Math. Soc. 350 (1998), 1–41.Google Scholar
[6] Kearnes, K. A., Kiss, E. W., and Valeriote, M. A., A geometric consequence of residual smallness. Ann. Pure Appl. Logic. 99 (1999), 137–169.Google Scholar
[7] Kearnes, K. A. and Willard, R. D., Inherently nonfinitely based solvable algebras. Canad. Math. Bull. 37 (1994), 514–521.Google Scholar
[8] Kiss, E. W., An easy way to minimal algebras. Internat. J. Algebra Comput. 7 (1997), 55–75.Google Scholar
[9] McKenzie, R., The residual bounds of finite algebras. Internat. J. Algebra Comput. 6 (1996), 1–28.Google Scholar
[10] Quackenbush, R. W., Equational classes generated by finite algebras. Algebra Universalis 1 (1971), 265–266.Google Scholar
[11] Shallon, C., Non-finitely based algebras derived from lattices. Ph.D. thesis, UCLA, 1978.Google Scholar
[12] Taylor, W., Subdirectly irreducible algebras in regular permutable varieties. Proc. Amer.Math. Soc. 75 (1979), 196–200.Google Scholar