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The word product in mathematics generally means the result of some type of multiplication operation, i.e. of certain types of mapping $X\!\times\!X\rightarrow Y$; 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 (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 connection 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 $n$.
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