![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
sniffnoyOK, just a quick followup to these entries: Loops with the inverse property are not enough to get gaps or well-ordering. You can iterate the "generalized dihedral loop" construction, and if the input is a loop with the inverse property, then so is the output. Meanwhile the fraction of involutions approaches 1.
So: It might work for groups. It might work for Moufang loops. (It definitely works for abelian groups.) But it doesn't work for monoids (you can't get rid of inverses) and it doesn't work for loops with the inverse property (you do need some sort of associativity). (Notionally, one could try it for Bol loops or Bruck loops? I am probably not going to do that.)
Meanwhile, over on the ring-like-things side of things, it currently seems entirely plausible to me that all this works even for rngs without associativity, i.e., having absolutely no constraints on the multiplication other than distributivity. Being built on top of an abelian group does a lot, I guess?
-Harry
So: It might work for groups. It might work for Moufang loops. (It definitely works for abelian groups.) But it doesn't work for monoids (you can't get rid of inverses) and it doesn't work for loops with the inverse property (you do need some sort of associativity). (Notionally, one could try it for Bol loops or Bruck loops? I am probably not going to do that.)
Meanwhile, over on the ring-like-things side of things, it currently seems entirely plausible to me that all this works even for rngs without associativity, i.e., having absolutely no constraints on the multiplication other than distributivity. Being built on top of an abelian group does a lot, I guess?
-Harry
