This might seem like a very stupid question and it probably is, but hopefully not as much as one could expect.
My question in fact is the following: Does the difference between sum and product lie only in the distinction of their respective neutral elements (i.e. "0" for addition and "1" for multiplication) and in the "one-way" distributive property of multiplication over addition?
As a newbie coming to this subject my first instinct while reading up on the basics of abstract algebra (AA from here on) was to take literally the "abstract" in AA, which — as I take it — doesn't really have to do with numbers per se, but "the structures arising from different ways of combining things together", of which numbers are a very useful and recurring instance, but not the only one.
For instance i have learned (from a 3b1b video on youtube) that the set of isomorphic rotations of a polygon is a group, since it is closed under addition, where addition is computed by applying one rotation after the other (also commutativity applies, there exists a "zero" element, and for every non-zero element there exists the additive inverse of that element). Here "addition" seemed to me to have nothing to do with the familiar algebraic summation of numbers, except for the properties that it holds.
Is this the right way of interpreting the subject?
That is, am i right in believing that the definition of an operation (a mapping of a set upon itself of the kind $S\times S \longrightarrow S$) exactly coincides with its properties and nothing more? (setting aside the fact that we can elaborate different methods of computing the output of these binary operations on different objects)