Chief Factor Sizes in Finitely Generated Varieties

Let A be a k-element algebra whose chief factor size is c. We show that if B is in the variety generated by A, then any abelian chief factor of B that is not strongly abelian has size at most ck−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 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.