Is there a fundamental difference in the things we call multiplication and those we call addition?
In a field, both binary operations obey exactly the same rules (commutativity, associativity, identity element, and inverse element [actually this one is the same for all but 1 element: namely $0$]). In a ring, some of the multiplicative rules of a field are relaxed, but it seems like we could just as easily have relaxed the additive rules.
It seems that even the distributive law could also be defined so that addition distributes over multiplication (opposite to the normal way), and thus is only a distinguishing property — when it even applies — because we've decided it should be.
Other than the fact that often we require both operations on whatever set of "numbers" we're considering, is there some property of addition that is never shared by multiplication (and vice versa), no matter which generalization of each we choose? That is, can we define the necessary and sufficient conditions for a binary operation to be called "addition" or "multiplication"?
Best Answer
The property that distinguishes addition from multiplication is the distributive law. This is part of the convention of calling the operations addition and multiplication. If the distributive law wasn't there, we wouldn't call the operations by those names.
The fact that multiplication distributes over addition admits $0\cdot a = 0 \;\forall a$ where $0$ is the additive identity. There is no such analog for addition. This is one property that falls out of imposing the distributive law.
Another necessary property is that if every element has an additive inverse, then addition must be commutative. Observe the commutativity of addition derived from the distributive law and the existence of additive inverses: $$\begin{align}(1+1)(a+b)\quad &= 1(a+b) + 1(a+b)\\ &= (1+1)a + (1+1)b\end{align}$$ $$a+b+a+b\quad =\quad a+a+b+b$$ $$b+a\quad=\quad a+b$$
Besides this, the addition and multiplication operations on a set $S$ are simply functions: $$\begin{align} +&\;:\;S\times S \rightarrow S\\ \cdot\:&\;:\;S\times S \rightarrow S \end{align}$$
They can be any mapping we choose so long as the distributive law is upheld.
The commutativity of multiplication can be relaxed, but as explained, commutativity of addition is required if every element has an additive inverse.