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| 2026-02-24 10:59:28 +0100 | <int-e> | tomsmeding: here is what it looks like properly: https://int-e.eu/~bf3/tmp/hapo.png |
| 2026-02-24 10:57:21 +0100 | <__monty__> | tomsmeding: Oh, obviously, darn, less interesting than I thought. |
| 2026-02-24 10:56:52 +0100 | <tomsmeding> | yes, which is also what dminuoso made of it |
| 2026-02-24 10:56:24 +0100 | <Leary> | tomsmeding: Cross connections. Bit hard in slapdash asci. I revised it with an X if that's clearer. |
| 2026-02-24 10:56:14 +0100 | <jreicher> | __monty__: if the order is transitive and antisymmetric it can't be cyclical |
| 2026-02-24 10:55:11 +0100 | <tomsmeding> | __monty__: Integer? |
| 2026-02-24 10:54:55 +0100 | <tomsmeding> | dminuoso: the x must be a crossing of lines without a node, yes |
| 2026-02-24 10:54:53 +0100 | <dminuoso> | Not sure if it makes it less or more ambiguous |
| 2026-02-24 10:54:51 +0100 | <merijn> | __monty__: Clearly we need a PartialOrd superclass of Ord that implement "a -> a -> Maybe Ordering" |
| 2026-02-24 10:54:43 +0100 | <dminuoso> | Or well, maybe without that x even |
| 2026-02-24 10:54:34 +0100 | <__monty__> | I'm also interested in kind of the reverse question, are there structures that can implement Ord but not Bounded? Something like a cyclical ordering? I suppose only if we put the "laws" of the class aside. |
| 2026-02-24 10:54:13 +0100 | <dminuoso> | Leary: Did you mean it like https://paste.tomsmeding.com/ayvscggm ? |
| 2026-02-24 10:53:19 +0100 | <int-e> | Hasse diagram, meh. |
| 2026-02-24 10:53:18 +0100 | <tomsmeding> | Leary: that notation in your gist is a bit ambiguous, what do the /\ on line 4 mean :p |
| 2026-02-24 10:52:55 +0100 | <jreicher> | int-e: have I understood you correctly? |
| 2026-02-24 10:52:51 +0100 | <int-e> | the translation is a bit lossy ;) |
| 2026-02-24 10:52:41 +0100 | <int-e> | tomsmeding: sorry, I'm trying to cast a Hesse diagram in ASCII here |
| 2026-02-24 10:52:37 +0100 | tomsmeding | closes the image editor |
| 2026-02-24 10:52:36 +0100 | <jreicher> | I think int-e and I are thinking the same. 0 is one end and 5 is the other. The join of 1 and 2 is not unique |
| 2026-02-24 10:52:27 +0100 | <tomsmeding> | yes perfect |
| 2026-02-24 10:52:14 +0100 | <tomsmeding> | yes |
| 2026-02-24 10:52:07 +0100 | <tomsmeding> | oh notation |
| 2026-02-24 10:52:00 +0100 | <int-e> | tomsmeding: it's transitive |
| 2026-02-24 10:51:53 +0100 | <tomsmeding> | int-e: no, because 2 !< 5 |
| 2026-02-24 10:51:47 +0100 | <Leary> | tomsmeding: https://gist.github.com/LSLeary/c52be6739f5815d13a6774d46dcbde75 |
| 2026-02-24 10:51:43 +0100 | <int-e> | tomsmeding: 5 is the unique maximal element here |
| 2026-02-24 10:51:42 +0100 | <tomsmeding> | ah :p |
| 2026-02-24 10:51:30 +0100 | <__monty__> | tomsmeding: The conversation has moved on but I'm not sure it's not leaving dminuoso behind ; ) |
| 2026-02-24 10:51:29 +0100 | <tomsmeding> | but there's no unique maximum |
| 2026-02-24 10:51:04 +0100 | <int-e> | Is it okay if it's? 0 < 1,2 < 3,4 < 5 has minimal and maximal elements but 3,4 have no infimum and 1,2 have no supremum |
| 2026-02-24 10:50:48 +0100 | <tomsmeding> | __monty__: yes, we figured that out, but we're already at the next question: that thing is clearly a lattice, but we want a not-lattice that is nevertheless a bounded partial order |
| 2026-02-24 10:50:30 +0100 | <jreicher> | Then just "duplicate" a meet or join. It will no longer be a lattice. |
| 2026-02-24 10:50:17 +0100 | <__monty__> | but not Ord. |
| 2026-02-24 10:50:11 +0100 | <__monty__> | dminuoso: If you're looking for any example of it whatsoever I think the one tomsmeding mentioned is the easiest. Take any set with more than one element and all subsets of that set, the partial order relation is subset-of, the minBound would be the empty set, the maxBound the set you started with, some pairs of subsets won't satisfy the subset-of relation in either order. So this can implement Bounded |
| 2026-02-24 10:50:02 +0100 | <tomsmeding> | or at least, I do |
| 2026-02-24 10:49:59 +0100 | <tomsmeding> | yes |
| 2026-02-24 10:49:54 +0100 | <jreicher> | tomsmeding: by "bounded" do we just mean the entire set (rather than any pair) as an infimum and supremum? |
| 2026-02-24 10:49:16 +0100 | <tomsmeding> | now we want an example of that :p |
| 2026-02-24 10:49:08 +0100 | <tomsmeding> | int-e: then we thought "but is Bounded then a bounded lattice?", to which the answer is no, as no unique joins/meets are required |
| 2026-02-24 10:49:03 +0100 | <int-e> | tomsmeding: Ah. I see, lack of context strikes again. |
| 2026-02-24 10:48:58 +0100 | <Leary> | It's not hard to make a counterexample. |
| 2026-02-24 10:48:45 +0100 | <Leary> | tomsmeding: You can just contrive a small finite partial order on paper. |
| 2026-02-24 10:48:44 +0100 | <tomsmeding> | int-e: the original original question was why Bounded does not have Ord as a superclass, which had answer "partial orders are a thing" |
| 2026-02-24 10:48:22 +0100 | <int-e> | I'm confused; Bounded just defined two points called minBound and maxBound? So it would work for partial orders too (and thus, lattices) |
| 2026-02-24 10:48:02 +0100 | <Leary> | dminuoso: Maximal means nothing else is bigger than it, though perhaps the /greatest/ element is a better notion for a bound, in which case its bigger than everything. |
| 2026-02-24 10:47:53 +0100 | <tomsmeding> | (I don't) |
| 2026-02-24 10:47:51 +0100 | fp | (~Thunderbi@wireless-86-50-141-0.open.aalto.fi) fp |
| 2026-02-24 10:47:46 +0100 | <tomsmeding> | do you have an example of a bounded partial order that is not a lattice? |
| 2026-02-24 10:47:29 +0100 | <tomsmeding> | jreicher: I think yes |
| 2026-02-24 10:47:24 +0100 | <tomsmeding> | (which are, in particular, subsets) |
