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 and has proved to be a powerful generalization of the notion of direct product. A subdirect product is a subalgebra (in the sense of universal algebra) A of a direct product ΠiAi such that every induced projection (the composite pjs: A → Aj of a projection pj: ΠiAi → Aj with the subalgebra inclusion s: A → ΠiAi) is surjective. A direct (subdirect) representation of an algebra A is a direct (subdirect) product isomorphic to A. An algebra is called 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. 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 (i.e. the whole direct power) 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.