Canonical examples of commutative loops

Question

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?

Answer 1

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.

Answer 2

If you haven't come across them already or simply moved on, check out commutative automorphic loops. Their inner mappings (whatever that means) are automorphic and every element of the loop oddly generates a group! There even seems to be a pretty rich culture around them with enough open questions for research. Also one could be missing a hypothesis (or 2 or 3 or... or n) but [3] mentions in passing that every group is actually a CA loop!! Moufang loops are also very extensively studied of which there exists a subset consisting of commutative ones.

References

[1]V. D. Belousov, Foundations of the Theory of Quasigroups and Loops

[2]R. H. Bruck and L. J. Paige, Loops whose inner mappings are automorphisms (the initiators of commutative A-loop theory)

[3]The structure of commutative automorphic loops https://arxiv.org/pdf/0810.1065.pdf (a fun introduction filled with sub loops, simple loops, Lagrange/Cauchy for loops, loop decompositions and more! (and it's all commutative!(although there is power associativity which luckily looks nothing like associativity)))

[4]Constructions of commutative automorphic groups https://pdfs.semanticscholar.org/19ba/26680fffeff652ba970f26429cc87275d718.pdf (perhaps one of the undoubtedly many papers that will be credited with the complete classification of loops with probably a lot of restrictions lol)

[5]Identities and relations in commutative Moufang loops https://ac.els-cdn.com/002186937490129X/1-s2.0-002186937490129X-main.pdf?_tid=941bf345-dc5b-4d95-9afa-18787c76de0e&acdnat=1525291386_3d26f67852aab92e3601ae6d88d0ca33 (Only for finitely generated though)

And to bring everything together with a taste of free, normal, and quotient loops:

[6]The structure of free automorphic Moufang loops http://www.ams.org/journals/proc/2012-140-07/S0002-9939-2011-11085-2/S0002-9939-2011-11085-2.pdf

I find it interesting how intrinsically groups are connected to loops although that makes sense seeing as how effectively groups encode symmetry and everything has at least some symmetry (dare I say super symmetry!?).

In response to why commutative loops aren't as fundamental as groups, technically they're more fundamental at least with respect to Abelian groups simply by virtue of having less axioms. While less axioms generally lead to a more pedagogical presentation, it doesn't necessarily lead to a more fruitful universe of ideas. Groups while more complex are simply more abundant in the mainstream of mathematics and easier to give concrete examples. Perhaps there will be a loop revolution and they'll replace groups as the intro to abstract algebra although if such a revolution was to occur I'd place my money on magmas.

All that being said, there is a very high probability (at least in my mind) that loops and all the other simpler algebraic systems possess a more fundamental form of symmetry. It may not be as "rich" as groups, rings whatev, but it certainly is key. I mean fields are made of rings and rings of groups (with some added fuzz) so why not keep descending until you get to the bare atoms of algebra!? Simplicity will always succeed in the end. It seems at first many ideas are thrown away due to their apparent "trivialness" although I believe in the words of Grothendieck, eventually that of loops and all more elementary algebraic structures will be "submerged and dissolved by some more or less vast theory, going well beyond the results originally to be established." Of course that's where universal algebra comes in (maybe... I recall reading a universal algebraist speaking of the limits of the theory and that perhaps the name is underserving). Then there will be a universal universal algebra (with a relabeling of title though) and then $universal^n$ algebra heck why stop there might as well make it a continuum and throw some morphisms in there and topologize and see what geometries you get in your non-stabilizing ascending continuous chain of algebraic theories describing "simpler" algebraic theories XD The rabbit hole never ends.

Anyway that's all, keep looping and God bless!