2 results
A Subalgebra Intersection Property for Congruence Distributive Varieties
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- Journal:
- Canadian Journal of Mathematics / Volume 61 / Issue 2 / 01 April 2009
- Published online by Cambridge University Press:
- 20 November 2018, pp. 451-464
- Print publication:
- 01 April 2009
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- Article
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We prove that if a finite algebra
$\mathbf{A}$ generates a congruence distributive variety, then the subalgebras of the powers of $\mathbf{A}$ satisfy a certain kind of intersection property that fails for finite idempotent algebras that locally exhibit affine or unary behaviour. We demonstrate a connection between this property and the constraint satisfaction problem.
Chief Factor Sizes in Finitely Generated Varieties
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- 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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- Article
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- You have access
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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.
