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)
Best Answer
This is subtle. You could say that the lesson of abstract algebra is that there is this more fundamental concept of an "operation," and both "addition" and "multiplication" are names we give to certain operations, first of all
For example we also add functions, random variables, matrices, etc. and we also multiply matrices, elements of an arbitrary group, etc.
Conventionally it's common to think of the group operation of an abelian group as "addition" and the group operation of a not-necessarily-abelian group as "multiplication" but strictly speaking it's not necessary to do this and we do this only to guide our intuition. If you wanted to be very rigorous and careful you could insist on only talking in terms of "operations" here.
Also conventionally when we are talking about a ring we call one of its operations "addition" and the other one "multiplication" and multiplication is the one that distributes over addition. (This convention is compatible with the previous one; a ring has an underlying abelian group given by its "addition," as well as a multiplicative not-necessarily-abelian group given by its "multiplication" on invertible elements.) These operations generalize addition and multiplication of numbers but may look very different; it would be very annoying to have to come up with completely new names for them in general so we just keep calling them addition and multiplication. As you learn the subject you learn which intuitions about ordinary addition and multiplication carry over to this more abstract setting and which don't.
A major reason we don't make hard and fast distinctions between "additions" and "multiplications" in abstract algebra is that there are (sometimes) homomorphisms relating them, so they're not as distinct as they might appear. For example the exponential $x \mapsto \exp(x)$ is a group homomorphism, and in fact an isomorphism, from the real numbers $\mathbb{R}$ under addition to the positive reals $\mathbb{R}_{>0}$ under multiplication, with inverse the logarithm! This is a fancy way of talking about the familiar exponent and logarithm rules $\exp(a + b) = \exp(a) \exp(b)$ and $\log(ab) = \log(a) + \log(b)$.