nLab
Mal'cev variety

A Mal’cev operation on a set $X$ is a ternary operation, a function

$$t:X\times X\times X\to X,(x,y,z)\mapsto t(x,y,z),$$t:X\times X\times X\to X,\,\,\,(x,y,z)\mapsto t(x,y,z) ,

which satisfies the identities $t(x,x,z)=z$ and $t(x,z,z)=x$. An important motivating example is the operation $t$ of a heap.

An algebraic theory $T$ is a Mal’cev theory when $T$ contains a Mal’cev operation. An algebraic theory is Mal’cev iff one of the following equivalent statements is true:

1. in the category of $T$-algebras, every internal reflexive relation is a congruence;

2. in the category of $T$-algebras, the composite (as internal relations) of any two congruences as a congruence;

3. in the category of $T$-algebras, the composition of equivalence relations is commutative.

Statement (i) is one of the motivations to introduce the notion of Mal'cev category.

A Mal’cev variety is the category of $T$-algebras for a Mal’cev theory $T$, thought of as a variety of algebras.

Mike Shulman: Surely you mean a variety of algebras, rather than an algebraic variety?

Toby: Yeah, I'm always doing that!

References

See the monograph Borceux-Bourn.

Spelling

The original is ‘Мальцев’; besides ‘Malʹcev’, this has also been transliterated ‘Malcev’ and ‘Maltsev’.