Pedagogically, when students are exposed to algebraic structures it seems standard for the major emphasis, if not all the emphasis, to be on groups, rings, R-modules, and categories. These are rich structures with interesting properties, but in the big picture, I have wondered why some defining properties make for a rich structure, while other properties gives less interesting structures, or nothing worth teaching at all.
As a motivating example, a set (or class, whatever) that is closed under some operation seems necessary to talk about anything meaningful; however, why is the particular combination of
- Having inverse elements
- Having an identity element
more rich (a group) than simply replacing associativity with commutativity (a structure I don't even know a name for)? I have also wondered why associativity is much more prevalent than commutativity. As another motivating example, we teach much about groups and rings but why not loops, monoids, semilattices, and near-rings? What makes the former set either richer in structure or more pedagogically sound to teach?
Even in category theory I can ask what makes the specific combination of defining properties of a category so great. —why associativity and not commutativity? —why categories and not semi categories? I wonder why its particular combination of defining properties is more "powerful", deep, and pervasive than another combination of properties.