Direct product

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In mathematics, one can often define a direct product of objects already known, giving a new one. This is generally 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, in others it is a different concept.

Contents

Examples

  • If we think of \mathbb{R} as the set of real numbers, then the direct product \mathbb{R}\times \mathbb{R} is precisely just the cartesian product, \{ (x,y) | x,y \in \mathbb{R} \}.
  • If we think of \mathbb{R} as the group of real numbers under addition, then the direct product \mathbb{R}\times \mathbb{R} still consists of \{ (x,y) | x,y \in \mathbb{R} \}. The difference between this and the preceding example is that \mathbb{R}\times \mathbb{R} is now a group. We have to also say how to add their elements. This is done by letting (a,b) + (c,d) = (a + c,b + d).
  • If we think of \mathbb{R} as the ring of real numbers, then the direct product \mathbb{R}\times \mathbb{R} again consists of \{ (x,y) | x,y \in \mathbb{R} \}. To make this a ring, we say how their elements are added, (a,b) + (c,d) = (a + c,b + d), and how they are multiplied (a,b)(c,d) = (ac,bd).

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