congruence_n-permutable - MathStructures

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=====Congruence n-permutability=====

An algebra is \emph{congruence $n$-permutable} if for all congruence relations $\theta,\phi$ of the algebra
$\theta\circ\phi\circ\theta\circ\phi\circ...=\phi\circ\theta\circ\phi\circ\theta\circ...$, where $n$ congruences
appear on each side of the equation.

A class of algebras is \emph{congruence $n$-permutable} if each of its members is congruence $n$-permutable. 

The term \emph{congruence permutable} is short for congruence $2$-permutable, i.e. $\theta\circ\phi=\phi\circ\theta$.

Congruence permutability holds for many 'classical' varieties such as groups, rings and vector spaces. 

Congruence $n$-permutability is characterized by a Mal'cev condition.

For $n=2$, a variety is congruence permutable iff there exists a term $p(x,y,z)$ such that the identities $p(x,z,z)=x=p(z,z,x)$ hold in
the variety.

=== Properties that imply congruence $n$-permutability ===

=== Properties implied by congruence $n$-permutability ===

Congruence $n$-permutability implies congruence $n+1$-permutability.

Congruence $3$-permutability implies congruence modularity[([Bjarni Jónsson, \emph{On the representation of lattices}, Math. Scand, \textbf{1}, 1953, 193-206 [[http://www.ams.org/mathscinet-getitem?mr=15:389d|MRreview]])].

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