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

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

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on vectors the scalar product^{}, the vector product, the dyad product and the Hadamard product^{}; on ideals the product of ideals;

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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  20130322 15:45:22 
Last modified on  20130322 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 