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sniffnoySo, I've written before -- both on my website and here -- about equational probabilities in groups. Beren Gunsolus recently posed to me the question, what if we generalized beyond groups?
Well, equational probabilities in monoids don't really seem to be interesting as best I can tell, or at least not interesting in this way. One can pretty easily find equations that have no gap; e.g., a²=1, or a²=a³ if one insists on equations that don't use 1. (I'll skip listing examples; they're not too hard to construct.)
But what if instead of getting rid of inverses, we weaken associativity? Say, loops? Or, to be a bit more restrictive, Moufang loops?
That equational probabilities in finite Moufang loops might exhibit gaps or well-ordering seems... kind of plausible. So -- I don't know much about Moufang loops, but like... apparently they satisfy Lagrange's Theorem, the order of the subloop divides the order of the loop?? Of course this is true for some complicated reason, not the easy reason it's true for groups, but it's apparently true. So that's a point in favor.
Another reason to look at something more restrictive than just loops is that -- so, in groups, we can look not only at the finite case, but also the compact case. Well, it turns out that locally compact inverse-property loops have Haar measure as well! So if we're looking at Moufang loops, or inverse-property loops more generally, we can again ask about the compact case, not just the finite case. I'm not going to go all the way out to inverse-property loops though; I'm going to keep things to Moufang loops, I mean those are hard enough!
So let's look at the simplest nontrivial cases from groups, x²=1 and xy=yx. (Another really interesting case to look at in Moufang loops would likely be associativity, x(yz)=(xy)z! But obviously that one is trivial in groups.) Do the elementary gap proofs (3/4 for x²=1 and 5/8 for xy=xy) carry over to the Moufang loop case? (Possibly with a weaker gap? In particular, it wouldn't surprise me if that 5/8 for commutativity in the group case becomes 2/3 or 3/4 in the Moufang loop case.)
As best as I can tell, the answer is no. The ordinary proofs for groups are all about looking at centers and centralizers. Now, in a Moufang loop, the commutant -- it's called the "commutant" rather than the "center", because the "center" refers to the intersection of the commutant with the nucleus, the nucleus being the subloop of all the elements that associate with everything -- is indeed a subloop. (Although it's apparently not always a normal subloop.) But as best I can tell, you don't get centralizers (commutalizers?) as subloops. So, as best I can tell, those proofs don't work.
(And even if these proofs do work, they might not extend to the compact case like the group proofs do... finite Moufang loops satisfy the Lagrange property, sure, but is the "index" meaningful? If you had a compact Moufang loops, would a subloop necessarily have measure 1/n or 0?)
...of course, maybe they are subloops, and it just doesn't have an easy proof! That does seem to be how Moufang loops seem to go, things are true but the proofs are hard. And, well, even if they're not subloops, a gap theorem could still hold for another reason!
So, it's an interesting question. I don't really intend to work on this much more, and this is even more speculative than the case of probabilities in groups, but it's certainly an interesting question...
-Harry
Well, equational probabilities in monoids don't really seem to be interesting as best I can tell, or at least not interesting in this way. One can pretty easily find equations that have no gap; e.g., a²=1, or a²=a³ if one insists on equations that don't use 1. (I'll skip listing examples; they're not too hard to construct.)
But what if instead of getting rid of inverses, we weaken associativity? Say, loops? Or, to be a bit more restrictive, Moufang loops?
That equational probabilities in finite Moufang loops might exhibit gaps or well-ordering seems... kind of plausible. So -- I don't know much about Moufang loops, but like... apparently they satisfy Lagrange's Theorem, the order of the subloop divides the order of the loop?? Of course this is true for some complicated reason, not the easy reason it's true for groups, but it's apparently true. So that's a point in favor.
Another reason to look at something more restrictive than just loops is that -- so, in groups, we can look not only at the finite case, but also the compact case. Well, it turns out that locally compact inverse-property loops have Haar measure as well! So if we're looking at Moufang loops, or inverse-property loops more generally, we can again ask about the compact case, not just the finite case. I'm not going to go all the way out to inverse-property loops though; I'm going to keep things to Moufang loops, I mean those are hard enough!
So let's look at the simplest nontrivial cases from groups, x²=1 and xy=yx. (Another really interesting case to look at in Moufang loops would likely be associativity, x(yz)=(xy)z! But obviously that one is trivial in groups.) Do the elementary gap proofs (3/4 for x²=1 and 5/8 for xy=xy) carry over to the Moufang loop case? (Possibly with a weaker gap? In particular, it wouldn't surprise me if that 5/8 for commutativity in the group case becomes 2/3 or 3/4 in the Moufang loop case.)
As best as I can tell, the answer is no. The ordinary proofs for groups are all about looking at centers and centralizers. Now, in a Moufang loop, the commutant -- it's called the "commutant" rather than the "center", because the "center" refers to the intersection of the commutant with the nucleus, the nucleus being the subloop of all the elements that associate with everything -- is indeed a subloop. (Although it's apparently not always a normal subloop.) But as best I can tell, you don't get centralizers (commutalizers?) as subloops. So, as best I can tell, those proofs don't work.
(And even if these proofs do work, they might not extend to the compact case like the group proofs do... finite Moufang loops satisfy the Lagrange property, sure, but is the "index" meaningful? If you had a compact Moufang loops, would a subloop necessarily have measure 1/n or 0?)
...of course, maybe they are subloops, and it just doesn't have an easy proof! That does seem to be how Moufang loops seem to go, things are true but the proofs are hard. And, well, even if they're not subloops, a gap theorem could still hold for another reason!
So, it's an interesting question. I don't really intend to work on this much more, and this is even more speculative than the case of probabilities in groups, but it's certainly an interesting question...
-Harry
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Date: 2021-11-04 01:29 am (UTC)no subject
Date: 2021-11-04 01:42 am (UTC)