An algebraic theory is a Mal’cev theory when contains a Mal’cev operation. An algebraic theory is Mal’cev iff one of the following equivalent statements is true:
in the category of -algebras, the composite (as internal relations) of any two congruences as a congruence;
in the category of -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 -algebras for a Mal’cev theory , thought of as a variety of algebras.
If is a binary relation on sets, write to say that . If , are -algebras, then is an internal relation in - if the conditions , and , for any -ary operation of , jointly imply .
The set-theoretic composite of two internal relations in - is also an internal relation, and the equality relation is always internal, so we may (and will) apply ordinary set-theoretic reasoning in our proofs below.
If is a Mal’cev theory, then any internal reflexive relation in - is an internal equivalence relation.
If is a Mal’cev operation and is any internal reflexive relation on a -algebra , then is transitive because given , we infer , and this together with and gives since is internal. Also is symmetric, because if , we infer , which together with and gives .
If every internal reflexive relation is an internal equivalence relation, then the composite of any two internal equivalence relations is also an internal equivalence relation.
The hypothesis is that internal reflexive relations and internal equivalence relations coincide. But (internal) reflexive relations are clearly closed under composition: .
If internal equivalence relations are closed under composition, then composition of internal equivalence relations is commutative.
If and are equivalence relations and so is , then
If composition of internal equivalence relations in - is commutative, then the theory has a Mal’cev operation .
According to the yoga of (Lawvere) algebraic theories, -ary operations are identified with elements of , the free -algebra on generators (more precisely, the Lawvere theory is the category opposite to the category of finitely generated free -algebras). Thus we must exhibit a suitable element of .
Let be the generators of , and let be the generators of . Let be the unique algebra map taking and to and to , and let be the unique algebra map taking to and and to . An operation is Mal’cev precisely when
Let be the equivalence relation on given by the kernel pair of , and let be the kernel pair of . Then and , so . Then, since composition of equivalence relations is assumed commutative, . This means there exists such that and , or that and . This completes the proof.
The theory of groups, where , is Mal’cev.
The theory of Heyting algebras, where
If is Mal’cev, and if is a morphism of algebraic theories, then is Mal’cev. From this point of view, the theory of groups is Mal’cev because the theory of heaps is Mal’cev, and the theory of Heyting algebras is Mal’cev because the theory of cartesian closed meet-semilattices is Mal’cev.
See also Mal'cev category.
In any finitely complete category, the intersection of two congruences (equivalence relations) on an object is a congruence, so that the set of equivalence relations is a meet-semilattice.
In a regular category such as a variety of algebras, where there is a sensible calculus of relations and relational composition, it is a simple matter to prove that if is closed under relational composition, then is the join in . For, if , then
while if in , then
In a regular category, if is closed under relational composition (equivalently, if composition of equivalence relations is commutative), then is a modular lattice.
The (poset-enriched) category of relations in a regular category is an allegory, and hence satisfies Freyd’s modular law
whenever , , are relations. As we have just seen, the hypothesis implies that joins in are given by composition (so is a lattice), and so for we have
Therefore, if , we have both
and also . Thus implies : the modular law is satisfied in .
If is a Mal’cev theory, then the lattice of congruences on any -algebra is a modular lattice.
A similar argument shows that congruence lattices for -algebras , for a Mal’cev theory, satisfy the following property (stronger than the modular property):
Freyd-Scedrov’s Categories, Allegories (2.157, pp. 206-207) gives the following argument: given relations , between sets, it is “easily verified” that
Then, under the assumption that equivalence relations internal to - commute (so that the join of equivalence relations on is their relational composite ), the Desarguesian axiom follows immediately.
See the monograph Borceux-Bourn.
The original is ‘Мальцев’; besides ‘Malʹcev’, this has also been transliterated ‘Malcev’ and ‘Maltsev’.