non-cartesian set product? Foremost, this question is asked from a point of a computer scientist undergrad, so please don't nag me for inconsistent notation, or lack of proper vocabulary.

Is there a concept in mathematics for a non-cartesian product of sets, i.e. a one which maps to a set of products of all elements from two sets? Let me define more formally:

$$
for\quad { S }_{ 1 },{ S }_{ 2 },{ S }_{ 3 }\subseteq \mathbb{R},\quad let\quad { f }: ({ S }_{ 1 },{ S }_{ 2 })\mapsto { S }_{ 3 }\\ so\quad that\quad {f}({ S }_{ 1 },{ S }_{ 2 })=\{ x|\exists a\in { S }_{ 1 },\exists b\in { S }_{ 2 },(ab=x)\} 
$$
For example: let ${ S }_{ 1 }=\{ 1, 2, 3\}$ and ${ S }_{ 2 }=\{ 4, 5\}$.
Therefore ${ S }_{ 1 }.{ S }_{ 2 }=\{ 1*4,\quad 1*5,\quad 2*4,\quad 2*5,\quad 3*4,\quad 3*5\} =\{ 4,5,8,10,12,15\}$

It would seem to me that this concept would be useful and widely used in mathematics, yet I have never stumbled upon it. Does this concept exist? If yes, where and how is it used? Or is it something that I inherently misunderstood about sets that prevents such concept to be useful?
The common notation for the element-wise product of two sets (which must both be subsets of some given magma, i.e., a set with an operation) is $AB$ or $A\cdot B$ or, if the operation is addition, $A+B$ (or any other symbol for the operation) and is defined as the set $\{a+b\mid a\in A,b\in B\}$ (suitably adapted to use the relevant operation symbol). It is commonly used throughout mathematics whenever convenient.
It's called a binary operation. A common application is Graph Theory, esp. the graph product.

Its similar to a Cartesian product, but this returns an ordered pair $(a,b)$ without defining an operation on $a$ and $b$.