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sniffnoySo continuing from last entry, I thought I'd see what are the biggest actual non-1 probabilities I can get for small Moufang loops (for involutions, commutativity, and associativity); so if there are indeed gaps, we can get an idea what they might be.
Searching for small finite Moufang loops quickly turned up this paper; according to it, most of the non-associative Moufang loops of small order are what I'm going to call "generalized dihedral Moufang loops" (the paper, oddly, doesn't seem to give them a name).
Basically, if you have a group G, you can define a Moufang loop L of order 2|G| by adjoining an element x and defining (for g,h∈G)
gh = gh
g(hx) = (hg)x
(gx)h = (gh-1)x
(gx)(hx) = h-1g.
You'll notice that if G is abelian, then L is just the generalized dihedral group G⋊C2. If G is nonabelian, however, then L is a nonassociative Moufang loop.
Because these are most of the ones of small order, I decided to focus on these, in the hopes that these would give me the best non-1 probabilities. Also, in the group case, we know that the best non-1 probabilities for a²=1 and for ab=ba both come from the dihedral group D8, so this seemed like a decent direction to try.
The results are interesting. First off, involutions and commutativity. These follow exactly the same formulas as they do in the group case... except since the input no longer has to be abelian, we can now get larger numbers out!
Specifically,
PL(a²=1) = 1/2 + 1/2 * PG(a²=1).
Picking G=D8, so PG(a²=1) = 3/4 (i.e. the maximum), we get PL(a²=1) = 7/8, which is what's better than what's possible for a group!
Again, this is the same formula as in the case where L is a group and G is abelian; but in that case our maximum non-1 input would be 1/2.
As for commutativity, we get
PL(ab=ba) = 1/4 * PG(ab=ba) + 3/4 * PG(a²=1).
So again picking G=D8, yielding probabilities of 5/8 and 3/4 respectively (i.e. the maxima once again), we get a commuting probability of 23/32, again better than the 5/8 that's possible for a group!
Once again, this is the same formula as where L is a group and G is abelian, just, now we can pick larger inputs.
Edit: Worth noting also, that when we look at the "commutalizers", we see that some of them are of order 12, and so not subloops. So that's why it's possible to do better than the 3/4 and 5/8 bounds, it would seem.
Finally, what about associativity? What do we get for that? Well, what we end up getting is
PL((ab)c=a(bc)) = 1/8 + 3/4 * PG(ab=ba) + 1/8 * PG(abc=cba).
(Note there's no parentheses on the equation abc=cba, as it's taking place in a group.)
Now, that last probability is a new one, but it's not a difficult one. It's pretty easy to adapt the usual 5/8 commutativity argument to this case and see that this probability also can never be more than 5/8 if the group is not abelian; and that 5/8 is once again achieved by D8. (Indeed, for a generalized dihedral group, this probability will always be equal to the commutation probability.)
So, plugging this in, we get an association probability of 9/16. So, it is possible to have a nonassociative Moufang loop where the probability that 3 random elements associate is at least 9/16.
Of course, this is just what's possible with these generalized dihedral Moufang loops! It's possible the real numbers could be higher... or that there could not be a gap at all. But hey, these could be the correct numbers, right? :) (Hell, I'd say there's a decent chance they are!) And they certainly constrain what's possible...
Edit again: Actually, with regard to associativity, this is not the best possible! If we look at the octonion Moufang loop of 16 elements, then if I've done my calculations correctly, this has an association probability of 43/64, which is bigger than 9/16. Of course, its commutation probability is much worse...
-Harry
Searching for small finite Moufang loops quickly turned up this paper; according to it, most of the non-associative Moufang loops of small order are what I'm going to call "generalized dihedral Moufang loops" (the paper, oddly, doesn't seem to give them a name).
Basically, if you have a group G, you can define a Moufang loop L of order 2|G| by adjoining an element x and defining (for g,h∈G)
gh = gh
g(hx) = (hg)x
(gx)h = (gh-1)x
(gx)(hx) = h-1g.
You'll notice that if G is abelian, then L is just the generalized dihedral group G⋊C2. If G is nonabelian, however, then L is a nonassociative Moufang loop.
Because these are most of the ones of small order, I decided to focus on these, in the hopes that these would give me the best non-1 probabilities. Also, in the group case, we know that the best non-1 probabilities for a²=1 and for ab=ba both come from the dihedral group D8, so this seemed like a decent direction to try.
The results are interesting. First off, involutions and commutativity. These follow exactly the same formulas as they do in the group case... except since the input no longer has to be abelian, we can now get larger numbers out!
Specifically,
PL(a²=1) = 1/2 + 1/2 * PG(a²=1).
Picking G=D8, so PG(a²=1) = 3/4 (i.e. the maximum), we get PL(a²=1) = 7/8, which is what's better than what's possible for a group!
Again, this is the same formula as in the case where L is a group and G is abelian; but in that case our maximum non-1 input would be 1/2.
As for commutativity, we get
PL(ab=ba) = 1/4 * PG(ab=ba) + 3/4 * PG(a²=1).
So again picking G=D8, yielding probabilities of 5/8 and 3/4 respectively (i.e. the maxima once again), we get a commuting probability of 23/32, again better than the 5/8 that's possible for a group!
Once again, this is the same formula as where L is a group and G is abelian, just, now we can pick larger inputs.
Edit: Worth noting also, that when we look at the "commutalizers", we see that some of them are of order 12, and so not subloops. So that's why it's possible to do better than the 3/4 and 5/8 bounds, it would seem.
Finally, what about associativity? What do we get for that? Well, what we end up getting is
PL((ab)c=a(bc)) = 1/8 + 3/4 * PG(ab=ba) + 1/8 * PG(abc=cba).
(Note there's no parentheses on the equation abc=cba, as it's taking place in a group.)
Now, that last probability is a new one, but it's not a difficult one. It's pretty easy to adapt the usual 5/8 commutativity argument to this case and see that this probability also can never be more than 5/8 if the group is not abelian; and that 5/8 is once again achieved by D8. (Indeed, for a generalized dihedral group, this probability will always be equal to the commutation probability.)
So, plugging this in, we get an association probability of 9/16. So, it is possible to have a nonassociative Moufang loop where the probability that 3 random elements associate is at least 9/16.
Of course, this is just what's possible with these generalized dihedral Moufang loops! It's possible the real numbers could be higher... or that there could not be a gap at all. But hey, these could be the correct numbers, right? :) (Hell, I'd say there's a decent chance they are!) And they certainly constrain what's possible...
Edit again: Actually, with regard to associativity, this is not the best possible! If we look at the octonion Moufang loop of 16 elements, then if I've done my calculations correctly, this has an association probability of 43/64, which is bigger than 9/16. Of course, its commutation probability is much worse...
-Harry
