The direct product is a construction of structures from smaller structures.
Specifally, if is an index set, and is a family of structures of the same species, the direct product of the family , denoted or simply is the Cartesian product of the sets , with coordinatewise relations.
One can form direct products of groups, rings, modules, topologies, partially ordered sets, and other structures. However, certain structures do not admit direct products, e.g., totally ordered sets and fields. For instance, if and are totally ordered sets, then the set is only a partially ordered set, for if are elements of and are elements of , then the two elements are incomparable. Similarly, if and are fields, and is a non-zero element of , then the non-zero element has no inverse, since 0 has no inverse; thus the direct product is a ring, but not a field.
The direct product should not be confused with the direct sum, though the two constructions coincide when only finitely many structures are involved.