Is there something as the quality of something that can be added? Like "addable" or "joinable".
For example, numbers can be added, sets can be added, vectors can be added. What is the name of that quality that makes them able to be "added" together ?
I would like to know also if the same exists for things that can me multiplied / divided.
Maybe that doesn't exist at all neither ?
Are you familiar with the terms "group", "ring", and "field"? I think this is what you are looking for.
There are also terms for related structures - e.g. semigroup, monoid, magma for one binary operation, and semiring, skew field for two binary operations.
In general, you may be interested in the examples section of https://en.wikipedia.org/wiki/Algebraic_structure.