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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: kearnes@euclid.colorado.edu
E. W. Kiss
Affiliation:
Eötvös University, Department of Algebra and Number Theory, Múzeum krt. 6–8, 1088 Budapest, Hungary, email: ewkiss@cs.elte.hu
Á. Szendrei
Affiliation:
Bolyai Institute, Aradi vértanúk tere 1, H-6720 Szeged, Hungary, email: A.Szendrei@math.u-szeged.hu
R. D. Willard
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, email: rdwillar@uwaterloo.ca
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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

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