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1. CJM 2002 (vol 54 pp. 736)

Kearnes, K. A.; Kiss, E. W.; Szendrei, Á.; Willard, R. D.
Chief Factor Sizes in Finitely Generated Varieties
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

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