Subdirect product
In mathematics, especially in the areas of abstract algebra known as universal algebra, group theory, ring theory, and module theory, a subdirect product is a subalgebra of a direct product that depends fully on all its factors without however necessarily being the whole direct product. The notion was introduced by Birkhoff in 1944, generalizing Emmy Noether's special case of the idea for Noetherian rings, and has proved to be a powerful generalization of the notion of direct product.
Definition
A subdirect product is a subalgebra A of a direct product ΠiAi such that every induced projection is surjective.A direct 'representation' of an algebra A is a direct product isomorphic to A.
An algebra is called Subdirectly [irreducible algebra|subdirectly irreducible] if it is not subdirectly representable by "simpler" algebras. Subdirect irreducibles are to subdirect product of algebras roughly as primes are to multiplication of integers.
Birkhoff proved that every algebra all of whose operations are of finite arity is isomorphic to a subdirect product of subdirectly irreducible algebras.
Examples
- Every permutation group is a sub-direct product of its restrictions to its orbits.
- Any distributive lattice L is subdirectly representable as a subalgebra of a direct power of the two-element distributive lattice. This can be viewed as an algebraic formulation of the representability of L as a set of sets closed under the binary operations of union and intersection, via the interpretation of the direct power itself as a power set. In the finite case such a representation is direct if and only if L is a complemented lattice, i.e. a Boolean algebra.
- The same holds for any semilattice when "semilattice" is substituted for "distributive lattice" and "subsemilattice" for "sublattice" throughout the preceding example. That is, every semilattice is representable as a subdirect power of the two-element semilattice.
- The chain of natural numbers together with infinity, as a Heyting algebra, is subdirectly representable as a subalgebra of the direct product of the finite linearly ordered Heyting algebras. The situation with other Heyting algebras is treated in further detail in the article on subdirect irreducibles.
- The group of integers under addition is subdirectly representable by any family of arbitrarily large finite cyclic groups. In this representation, 0 is the sequence of identity elements of the representing groups, 1 is a sequence of generators chosen from the appropriate group, and integer addition and negation are the corresponding group operations in each group applied coordinate-wise. The representation is faithful because of the size requirement, and the projections are onto because every coordinate eventually exhausts its group.
- Every vector space over a given field is subdirectly representable by the one-dimensional space over that field, with the finite-dimensional spaces being directly representable in this way.
- Subdirect products are used to represent many small perfect groups in.
- Every reduced commutative Noetherian ring is a sub-direct product of integral domains. And more generally every commutative Noetherian ring is a sub-direct product of rings whose only zero-divisors are nilpotent.
- Every commutative reduced ring is a sub-direct product of fields.