Congruence-permutable algebra

In universal algebra, two congruences $\alpha ,\beta$ are said to be permutable if $\alpha \circ \beta =\beta \circ \alpha$ . A congruence-permutable algebra is one in which all congruences are permutable. A variety of algebras is congruence-permutable if every algebra in the variety is congruence-permutable.

A Mal'cev term is a term M for which

M(x,x,y)=y=M(y,x,x).

A congruence-permutable variety is one whose signature contains a Mal'cev term.

Most classical varieties in abstract algebra, such as groups, rings, and Lie algebras, are congruence-permutable. Any variety that contains a group operation is congruence-permutable, and the Mal'cev term is $xy^{-1}z$ .

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