Logs: freenode/#haskell
| 2021-03-22 17:42:19 | <ski> | (very loosely and a bit superficially inspired by "in case this makes you wonder, there *are* abelian infinite groups A with A =~ A^2 (and hence to A^n for all positive), such as the additive group of Z[x], or R (the reals). Way more bizarrely than that, there's an abelian torsionfree A with A =~ A^3 but not =~ A^2. I still need to go back to that paper and unravel all the details, including in an earlier |
| 2021-03-22 17:42:25 | <ski> | one it references.") |
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| 2021-03-22 18:10:42 | <tomsmeding> | ski: What's the isomorphism R^2 =~ R? |
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| 2021-03-22 18:11:43 | <ski> | i didn't ask (but i suppose the one they had in mind was non-continuous, incomputable) |
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| 2021-03-22 18:12:42 | <wroathe> | ghcjs is kind of a funny project isn't it... A seemingly huge amount of effort has gone into building it, but documentation and distribution methods seem to be nonexistant |
| 2021-03-22 18:12:53 | <pjb> | tomsmeding: https://en.wikipedia.org/wiki/Space-filling_curve |
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| 2021-03-22 18:17:47 | <tomsmeding> | pjb: so let I(x) = atan(2pi x - pi), and let [0,1] be a group with x + y = I^-1(I(x) + I(y)), and then for a space-filling curve f : [0,1] -> [0,1]^2 let [0,1]^2 be a group with (a,b) + (x,y) = f(f^-1((a,b)) + f^-1((x,y))) ? |
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| 2021-03-22 18:18:42 | <tomsmeding> | and then let R^2 be a group with (a,b) + (x,y) = (let (u,v) = (I^-1(a),I^-1(b)) + (I^-1(x),I^-1(y)) in (I(u),I(v))) |
| 2021-03-22 18:18:54 | <tomsmeding> | that's the weirdest way ever to claim that R =~ R^2 :p |
| 2021-03-22 18:19:01 | <tomsmeding> | or I'm missing something simpler |
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| 2021-03-22 18:19:24 | <pjb> | tomsmeding: yes, something like that. But if you limit yourself to a square, it's easier. You may not need a function as complex as a space filling function. |
| 2021-03-22 18:19:28 | <tomsmeding> | ('I' is just an arbitrary bijection between [0,1] and R) |
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| 2021-03-22 18:20:18 | <tomsmeding> | pjb: yeah if you forget about R and just talk about the unit {interval,square}, you can ditch that whole atan() nonsense |
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| 2021-03-22 18:20:29 | <tomsmeding> | but you still have the space-filling curve then, right? |
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| 2021-03-22 18:20:59 | <tomsmeding> | if someone uses "R^2" as a group, I'm not usually assuming they mean via a space-filling curve :p |
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| 2021-03-22 18:21:19 | <tomsmeding> | along that reasoning, R is group-isomorphic to any set of the same cardinalit |
| 2021-03-22 18:21:20 | <tomsmeding> | -y |
| 2021-03-22 18:21:36 | <tomsmeding> | anyway, tangent |
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| 2021-03-22 18:23:52 | <ski> | monochrom : "Mathematicians have the best designs on write-only notation." -- ooc, what was this apropos ? |
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| 2021-03-22 18:25:01 | <tomsmeding> | ski: https://ircbrowse.tomsmeding.com/selection/haskell?title=Conversation&events=396546,396547 |
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| 2021-03-22 18:25:53 | <tomsmeding> | ski: forgive me for spamming; what is a "least restrictive type"? |
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| 2021-03-22 18:27:51 | ski | noticed no spamming |
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| 2021-03-22 18:28:40 | <pjb> | tomsmeding: well, intuitively, I would want to do something like with integers, using the quotient and the remainder, but it may not work with reals. So a space filling curve might still be needed on a finite interval. |
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| 2021-03-22 18:28:45 | <ski> | oh, i guess you mean that was the context. i didn't connect them, when reading them |
| 2021-03-22 18:29:11 | <tomsmeding> | ski: yes I meant that was the context :) |
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| 2021-03-22 18:29:35 | <pjb> | Notably, there are as many reals between ]0,1[ as in the full ℝ set. So space-filling is most probably indicated. |
| 2021-03-22 18:29:58 | <tomsmeding> | blagh replace all uses of [0,1] with ]0,1[ in my messages just now |
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| 2021-03-22 18:30:30 | <pjb> | |ℝ| = |ℝ²| = 2^ℵ₀ |
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| 2021-03-22 18:31:48 | <tomsmeding> | I guess my point is that the claim was "there are abelian infinite groups A with A =~ A^2 (and hence to A^n for all positive n)" |
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| 2021-03-22 18:32:14 | <tomsmeding> | but the concept of the product of groups is well-defined, and uses (a,b) + (x,y) = (a+x, b+y), and not some kind of space-filling curve |
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| 2021-03-22 18:32:30 | <ski> | tomsmeding : the idea being (a) there's no "junk" (no extra things in the type/set, than what can be generated from the given operations and initial generators); and (b) no "confusion" (no extra identities/equalities/laws hold between elements, apart from those that are required/imposed) |
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| 2021-03-22 18:32:50 | <tomsmeding> | I see |
| 2021-03-22 18:33:32 | <tomsmeding> | and "satisfying 'T -> U'" means that that function T -> U must be injective, I guess? |
| 2021-03-22 18:33:36 | <ski> | tomsmeding : so, e.g. given a type `T', the type of (finite) lists on `T' is the least (restrictive) type which is a monoid, and which includes `T' (has a morphism/function from `T' to it) |
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