Function mapping notation for a binary operation on a set

Question

I found the following definition of a binary operation on a set from here:

A binary operation $*$ on a set $S$ is a map $\ast:S\times S \rightarrow S$

My question is, if I define an operation $\ast$, then should it be possible to reach every point in $S$ with some input in $S\times S$? What if I define an operation on a set for which the image was always in $S$, but could never be some parts of $S$?

The notation seems to suggest that this would not classify as a binary operation.

Answer 1

Defining such an operation would be fine. It just depends on how useful you want your binary operation to be. A binary operation that reaches every element of the set is an essential part of a mathematical group, which is a very important area of study. But there are objects called semi-groups that may not have this property, and are still useful.

Answer 2

You can certainly have binary operations on a set where not every element of the set is an output. For instance, addition on the positive integers is a lovely binary operation on the positive integers, but you can't reach $1$ as an output.