Direct product

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The direct product is a construction of structures from smaller structures.

Specifally, if $I$ is an index set, and $(A_i))_{i\in I}$ is a family of structures of the same species, the direct product of the family $(A_{i})_{i\in i}$, denoted \[\bigotimes_{i\in I} A_i ,\] or simply \[\prod_{i\in I} A_i,\] is the Cartesian product of the sets $A_i$, 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 $A$ and $B$ are totally ordered sets, then the set $A\times B$ is only a partially ordered set, for if $a_1 <a_2$ are elements of $A$ and $b_1 < b_2$ are elements of $B$, then the two elements \[(a_1,b_2), (a_2,b_1)\] are incomparable. Similarly, if $E$ and $F$ are fields, and $a$ is a non-zero element of $E$, then the non-zero element $(a,0)$ has no inverse, since 0 has no inverse; thus the direct product $E\times F$ 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.

See also