product

The word product  in mathematics generally means the result of some of multiplication operation, i.e. (http://planetmath.org/Ie) of certain of mapping  $X\times X\to Y$; such operations are commonly distributive over the addition operation of $X$ if it is defined.

If $x_1$ and $x_2$ are two elements of the set $X$, giving the product  $y\in Y$, then $x_1$ and $x_2$ are in general called the factors of this product.

Some most usual products are

• the ring product (http://planetmath.org/Ring) (also in fields), especially the product of numbers and the product of square matrices;

• on vectors the scalar product, the vector product, the dyad product and the Hadamard product; on ideals the product of ideals;

• the Cartesian product, the direct products of various systems (not in with any additions).
  

Such kinds of product that are associative, allow to form a product of more than two factors, which is justified by the theorem in the entry general associativity.  E.g. the usual product of the integers from 1 to $n$ is the factorial of $n$.

Title	product
Canonical name	Product
Date of creation	2013-03-22 15:45:22
Last modified on	2013-03-22 15:45:22
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	10
Author	pahio (2872)
Entry type	Definition
Classification	msc 03E20
Related topic	Multiplication
Related topic	RuleOfProduct
Related topic	FrobeniusProduct
Defines	factor
Defines	factor of a product