The compact MV-algebra problem
Feb. 5th, 2022 09:11 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
sniffnoySo! I've made a whole bunch of updates to my equational probabilities page. As a result I've eliinated the "???" category, because I managed to rule out nearly everything in that category; the exception, exponential rings, I decided to reclassify as "plausible??".
You can see the page for the details of all the things I've now ruled out that I previously hadn't. I do like how I managed to rule out both sloops and squags by switching betwenen them. :) But the most interesting case is the case of MV-algebras.
Now, there are two other cases on there -- near-rings with identity, and Kleene algebras in the lattice theory sense -- that I classified as "false for well-ordering", because I could rule out well-ordering, but couldn't rule out gaps; but in those cases I assume that the failure of well-ordering means gaps are unlikely. Whereas in the case of MV-algebras, I can prove that well-ordering fails, but gaps hold!
Or at least, they do in the finite case. And that's what I really want to discuss -- is there a compact case to speak of? By which I mean, can we meaningfully define probability in this case? Now of course, any probability measure we define won't be translation-invariant, but that doesn't mean there's no natural way to define one.
For instance: It turns out that MV-algebras are equivalent to lattice-ordered abelian group together with a designated nonnegative element (that also has to meet certain conditions if we want to keep things canonical). However, does that equivalence carry over when working over the category of topological spaces, rather than sets? I found a paper saying it carries over to topoi, but topological spaces don't form a topos, so that's no good.
Well, in truth, I don't particularly care about the answer to that question per se; I just care if we can canonically identify MV-algebras with lattice-ordered abelian groups with a designated nonegative element, I don't particularly care if this actually forms an equivalence of categories or not. So y'know I want the two constructions to be inverse to each other, but if they're not actually functorial, that's fine by me?
Anyway, the idea is that if this works, then perhaps a locally compact Hausdorff MV-algebra, and in particular a compact one, could be shown to necessarily come from a locally compact Hausdorff group, and necessarily has positive Haar measure (obviously it has to be finite measure since it's compact), and so we can get a probability measure on the MV-algebra that way.
...unfortunately, uh, there's a lot that would need to be proven to make that work. Topological spaces, and in particular quotients of topological spaces, can be nasty. (One step of the construction for going from MV-algebra to lattice-ordered abelian group requires taking a Grothendieck group, and that means a quotient.) Um, at least if either is Hausdorff than the other is, that I can show. And I think I can show that going from group to MV-algebra is left-inverse to going from MV-algebra to group (that is, it still is once topology is involved, since after all it's already known to be in the usual non-topological case!). But is it right-inverse? And what about the local compactness issue...?
Yeah, this has been bugging me. I might just not think too much more about this, since this is getting a bit far afield from the original question -- even if the infinite case is meaningful, I don't see how one would prove gaps there, or even have any idea if gaps would hold -- but maybe I should ask about this on MathOverflow...?
-Harry
You can see the page for the details of all the things I've now ruled out that I previously hadn't. I do like how I managed to rule out both sloops and squags by switching betwenen them. :) But the most interesting case is the case of MV-algebras.
Now, there are two other cases on there -- near-rings with identity, and Kleene algebras in the lattice theory sense -- that I classified as "false for well-ordering", because I could rule out well-ordering, but couldn't rule out gaps; but in those cases I assume that the failure of well-ordering means gaps are unlikely. Whereas in the case of MV-algebras, I can prove that well-ordering fails, but gaps hold!
Or at least, they do in the finite case. And that's what I really want to discuss -- is there a compact case to speak of? By which I mean, can we meaningfully define probability in this case? Now of course, any probability measure we define won't be translation-invariant, but that doesn't mean there's no natural way to define one.
For instance: It turns out that MV-algebras are equivalent to lattice-ordered abelian group together with a designated nonnegative element (that also has to meet certain conditions if we want to keep things canonical). However, does that equivalence carry over when working over the category of topological spaces, rather than sets? I found a paper saying it carries over to topoi, but topological spaces don't form a topos, so that's no good.
Well, in truth, I don't particularly care about the answer to that question per se; I just care if we can canonically identify MV-algebras with lattice-ordered abelian groups with a designated nonegative element, I don't particularly care if this actually forms an equivalence of categories or not. So y'know I want the two constructions to be inverse to each other, but if they're not actually functorial, that's fine by me?
Anyway, the idea is that if this works, then perhaps a locally compact Hausdorff MV-algebra, and in particular a compact one, could be shown to necessarily come from a locally compact Hausdorff group, and necessarily has positive Haar measure (obviously it has to be finite measure since it's compact), and so we can get a probability measure on the MV-algebra that way.
...unfortunately, uh, there's a lot that would need to be proven to make that work. Topological spaces, and in particular quotients of topological spaces, can be nasty. (One step of the construction for going from MV-algebra to lattice-ordered abelian group requires taking a Grothendieck group, and that means a quotient.) Um, at least if either is Hausdorff than the other is, that I can show. And I think I can show that going from group to MV-algebra is left-inverse to going from MV-algebra to group (that is, it still is once topology is involved, since after all it's already known to be in the usual non-topological case!). But is it right-inverse? And what about the local compactness issue...?
Yeah, this has been bugging me. I might just not think too much more about this, since this is getting a bit far afield from the original question -- even if the infinite case is meaningful, I don't see how one would prove gaps there, or even have any idea if gaps would hold -- but maybe I should ask about this on MathOverflow...?
-Harry
