I'm teaching myself abstract algebra.
We can define a magma $\left(M,\cdot\right)$ as $a,b\in M\implies a\cdot b\in M$.
I get confused when we talk about quasigroups.
A group must satisfy all conditions of a magma, therefore it must be a set and a single binary operation.
And thus, a quasigroup must have a single binary operation.
However, Wikipedia says,
Quasigroup: A magma where division is always possible.
Doesn't this imply that a quasigroup has two binary operations?
Or perhaps I am misunderstanding the meaning of division.
Does that mean that $a,b\in Q$ but not necessarily $a\div b\in Q$?
I think of the group of the set of integers with addition, $\left(\mathbb{Z},+\right)$.
If my assumption is correct, then this group satisfies the conditions of a quasigroup.
It satisfies $a,b\in G\implies a+b\in G$.
And it satisfies the condition that any element can divide another but not necessarily $a\div b\in G$.
Best Answer
As you are teaching yourself abstract algebra, you have noticed that there are things you must learn not take for granted. Literally you must subtract things that you thought you knew or that you associated with them.
Mathematicians have started with ideas born from the usual numbers and have looked if there was other instances of them.
But most teachers have a tendency to present certain facts or shortcuts or traditions as permanent and evident laws and they evaluate pupils on them when they are just consequences or special cases only valid in the context of what they teach.
Saying "addition, subtraction, multiplication, division are the four operations" is a good example. Even presenting them in this order is a choice. Subtraction can be seen as just an inverse operation of addition or something more natural and direct than addition in a concrete context : if you break a twig that you found in the forest, you are subtracting a part from it. Adding the part back together looks like a very abstract idea. Division is usually presented as a separate operation but is it ? Can you have two different division concepts for the same numbers with the same multiplication ?
You may define the same object or concept with different points of view.
As seen in comments, the division idea used in the definition you have seen is just one way to obtain or prescribe exclusive left and right divisors and prepare for the existence of neutral element.
Another look is the following:
In a quasigroup you can say that, at start, there is essentially one ternary relation, such that for each ordered couple of element, exactly one third element satisfies the relation.
expresses this relation (as R(a,b,c) would), as
and
would do as well.
In fact, each of these "operations" could be the unique operation of the quasigroup, having the others as right and left division.