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1. CJM 2011 (vol 65 pp. 3)

Barto, Libor
Finitely Related Algebras in Congruence Distributive Varieties Have Near Unanimity Terms
We show that every finite, finitely related algebra in a congruence distributive variety has a near unanimity term operation. As a consequence we solve the near unanimity problem for relational structures: it is decidable whether a given finite set of relations on a finite set admits a compatible near unanimity operation. This consequence also implies that it is decidable whether a given finite constraint language defines a constraint satisfaction problem of bounded strict width.

Keywords:congruence distributive variety, Jónsson operations, near unanimity operation, finitely related algebra, constraint satisfaction problem
Categories:08B05, 08B10

2. CJM 2009 (vol 61 pp. 451)

Valeriote, Matthew A.
A Subalgebra Intersection Property for Congruence Distributive Varieties
We prove that if a finite algebra $\m a$ generates a congruence distributive variety, then the subalgebras of the powers of $\m a$ 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.

Keywords:congruence distributive, constraint satisfaction problem, tame congruence theory, \jon terms, Mal'cev condition
Categories:08B10, 68Q25, 08B05

3. 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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