Subdirect product

Definition

Examples

• 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.