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 $n$.