Probabilities in rings??
Nov. 15th, 2021 10:47 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
sniffnoyFollowing up on this post, Drake asked me, what about probabilities in rings? (Or commutative rings?)
I initially dismissed this as not worth thinking about, but... it seems there could be something there? I dunno, it's hard to say. I certainly haven't found any counterexamples for commutative rings or rings more generally (i.e., any equations for which no gap exists, or more generally for which an infinite ascending chain exists), but I feel like mostly the only cases where I can show a gap *does* exist are cases where a gap exists for, like, stupid reasons. :P
The one nonstupid case I know of, in the not-necessarily-commutative setting, is once again commuting probabilities. It seems those once again have a 5/8 gap, for an exactly analogous reason -- I didn't come up with this myself, I found a paper stating the 5/8 gap and then noticed the proof was just the same as in the group case. Of course, if you want an example to show that 5/8 is achieved, that is different!
(EDIT January 1: Actually, I should note that one of the cases I considered "stupid" above is actually somewhat interesting -- considering x²=x in commutative rings.)
I had been kind of hoping there was a classification of finite commutative rings that I could go by, for the commutative setting, but it seems there isn't. The fact that finite rings are necessarily Artin rings of course tells you a lot immediately, but, finite commutative local rings can vary a fair bit, it seems. Oh well. (Of course, if you want to generalize to compact rings, those don't have to be Artin, so...)
I should note that if you try to generalize to semirings (or commutative semirings), then you don't get gaps; you can show that x+y=xy has no gap there, for instance.
So, hm. Probably not really going to go anywhere with this, but, I guess there may be something here after all?
(I'm kind of wondering now whether I should in fact attempt to generalize beyond Moufang loops, to inverse-property loops or something, but, uh, I don't really have a good idea of how I'd get any sense of that... IDK, I'll probably make some little attempt at this later to see what I can find but I don't expect to find much.)
-Harry
I initially dismissed this as not worth thinking about, but... it seems there could be something there? I dunno, it's hard to say. I certainly haven't found any counterexamples for commutative rings or rings more generally (i.e., any equations for which no gap exists, or more generally for which an infinite ascending chain exists), but I feel like mostly the only cases where I can show a gap *does* exist are cases where a gap exists for, like, stupid reasons. :P
The one nonstupid case I know of, in the not-necessarily-commutative setting, is once again commuting probabilities. It seems those once again have a 5/8 gap, for an exactly analogous reason -- I didn't come up with this myself, I found a paper stating the 5/8 gap and then noticed the proof was just the same as in the group case. Of course, if you want an example to show that 5/8 is achieved, that is different!
(EDIT January 1: Actually, I should note that one of the cases I considered "stupid" above is actually somewhat interesting -- considering x²=x in commutative rings.)
I had been kind of hoping there was a classification of finite commutative rings that I could go by, for the commutative setting, but it seems there isn't. The fact that finite rings are necessarily Artin rings of course tells you a lot immediately, but, finite commutative local rings can vary a fair bit, it seems. Oh well. (Of course, if you want to generalize to compact rings, those don't have to be Artin, so...)
I should note that if you try to generalize to semirings (or commutative semirings), then you don't get gaps; you can show that x+y=xy has no gap there, for instance.
So, hm. Probably not really going to go anywhere with this, but, I guess there may be something here after all?
(I'm kind of wondering now whether I should in fact attempt to generalize beyond Moufang loops, to inverse-property loops or something, but, uh, I don't really have a good idea of how I'd get any sense of that... IDK, I'll probably make some little attempt at this later to see what I can find but I don't expect to find much.)
-Harry
