Direct product

In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one talks about the product in category theory, which formalizes these notions. In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one talks about the product in category theory, which formalizes these notions. Examples are the product of sets (see Cartesian product), groups (described below), the product of rings and of other algebraic structures. The product of topological spaces is another instance. There is also the direct sum – in some areas this is used interchangeably, while in others it is a different concept. In a similar manner, we can talk about the direct product of finitely many algebraic structures, e.g. R × R × R × R {displaystyle mathbb {R} imes mathbb {R} imes mathbb {R} imes mathbb {R} } . This relies on the fact that the direct product is associative up to isomorphism. That is, ( A × B ) × C ≅ A × ( B × C ) {displaystyle (A imes B) imes Ccong A imes (B imes C)} for any algebraic structures A {displaystyle A} , B {displaystyle B} , and C {displaystyle C} of the same kind. The direct sum is also commutative up to isomorphism, i.e. A × B ≅ B × A {displaystyle A imes Bcong B imes A} for any algebraic structures A {displaystyle A} and B {displaystyle B} of the same kind. We can even talk about the direct product of infinitely many algebraic structures; for example we can take the direct product of countably many copies of R {displaystyle mathbb {R} } , which we write as R × R × R × ⋯ {displaystyle mathbb {R} imes mathbb {R} imes mathbb {R} imes dotsb } . In group theory one can define the direct product of two groups (G, ∘) and (H, ∙), denoted by G × H. For abelian groups which are written additively, it may also be called the direct sum of two groups, denoted by G ⊕ H {displaystyle Goplus H} .