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2 results

  • EVERY SHIFT AUTOMORPHISM VARIETY HAS AN INFINITE SUBDIRECTLY IRREDUCIBLE MEMBER

  • Part of
  • KATE S. OWENS
  • Journal:
    Journal of the Australian Mathematical Society / Volume 88 / Issue 2 / April 2010
    Published online by Cambridge University Press:
    19 January 2010, pp. 231-238
    Print publication:
    April 2010
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  • A shift automorphism algebra is one satisfying the conditions of the shift automorphism theorem, and a shift automorphism variety is a variety generated by a shift automorphism algebra. In this paper, we show that every shift automorphism variety contains a countably infinite subdirectly irreducible algebra.

  • Chief Factor Sizes in Finitely Generated Varieties

  • K. A. Kearnes, E. W. Kiss, Á. Szendrei, R. D. Willard
  • Journal:
    Canadian Journal of Mathematics / Volume 54 / Issue 4 / 01 August 2002
    Published online by Cambridge University Press:
    20 November 2018, pp. 736-756
    Print publication:
    01 August 2002
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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.