In most (but not all) cases where you name an operation multiplication is when you have two binary operations, and one of them distributes over the other, but not the other way round. Then the operation that distributes over the other is called multiplication and the other one addition, in analogy to the real numbers where multiplication distributes over addition.
Given that in those cases, almost always the addition is commutative, but often the multiplication isn't, there is also another convention where if you have only one operation, this operation is usually called multiplication if it doesn't need to be commutative, and addition if it is required to be commutative. For example, in general groups, the group operation is called multiplication, while in abelian groups, it is usually called addition.
Moreover in most cases where an operation is called multiplication, multiplication of natural or of real numbers is a special case of that structure.
For example, multiplication of ordinal numbers is not commutative, but the natural numbers are a subset of the ordinal numbers, and the restriction of ordinal multiplication to those gives the multiplication of natural numbers (and therefore this restriction is also commutative).
Also multiplication of real or complex matrices reduces to the multiplication of real or complex numbers in the case of $1\times 1$ matrices (or more generally, in the case of multiples of the identity matrix).
Also multiplication of non-zero real numbers is a specific instance of group multiplication, and without the restriction to non-zero numbers, a specific instance of field multiplication.
The cross product of vectors does not have such a special case, but it clearly distributes over addition.
celtschk
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2021-02-14T19:56:50Z (almost 4 years ago)