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# Chief Factor Sizes in Finitely Generated Varieties

Published:2002-08-01
Printed: Aug 2002
• K. A. Kearnes
• E. W. Kiss
• Á. Szendrei
• R. D. Willard
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## Abstract

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 {\it 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 $\mathbf{1}$. As a generalization, we bound the size of multitraces of types~$\mathbf{1}$, $\mathbf{2}$, and $\mathbf{3}$ by extending coordinatization theory. Finally, we exhibit some examples of bad behavior, even in varieties satisfying a congruence identity.
 Keywords: tame congruence theory, chief factor, multitrace
 MSC Classifications: 08B26 - Subdirect products and subdirect irreducibility