The context of this question is that I'm trying to understand why groups are so fundamental in modern math (a different question that has already been asked elsewhere).
So I was wondering: what if we swapped the associativity property of groups with the commutativity property. This gives you commutative loops (except that the left and right inverse need not be the same. We could alternatively also require this to be the case). Why isn't this structure studied as much as groups are, and why isn't it as fundamental?
So in order to defend commutative loops in its losing competition against groups, what are some canonical examples of commutative loops that are worth studying?
There are probably more reasons than one, but I always imagine that this is a important consideration: With associativity, there is the possibility of representing the elements of the set using functions. Functional composition is associative. This is what we are doing when we represent groups using matrix representations.
Representing things this way makes them a lot more concrete, and in a lot of fields you are encouraged to think using such constructs.
If you are going to defend loops, I don't see point of including commutativity. It looks like you were just looking for some random substitute for associativity. They are not really fit to be interchanged. I imagine that a great deal of interesting loops are noncommutative, so focusing on loops in general is probably the way to go.
Planar ternary rings are my favorite loops.