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| 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) |
| 2026-02-24 10:47:18 +0100 | <jreicher> | (But I'm still not sure what the initial question was) |
| 2026-02-24 10:47:11 +0100 | <jreicher> | Yes |
| 2026-02-24 10:47:06 +0100 | <dminuoso> | The uniqueness is not of any subset, but a *pair* of elements. |
| 2026-02-24 10:46:52 +0100 | <jreicher> | dminuoso: are you wondering what the difference between lattice and having max and min is? |
| 2026-02-24 10:46:44 +0100 | <dminuoso> | Hold on what I just said makes no sense. |
| 2026-02-24 10:46:10 +0100 | <jreicher> | tomsmeding: clopen is my favourite |
| 2026-02-24 10:45:35 +0100 | <dminuoso> | The favourite person in a group could work I suppose. |
| 2026-02-24 10:45:27 +0100 | <dminuoso> | Oh hold on. |
| 2026-02-24 10:45:20 +0100 | <tomsmeding> | welcome to math where there are unintuitive counterexamples |
| 2026-02-24 10:45:08 +0100 | <dminuoso> | tomsmeding: Trying to understand what "maximal element of the entire set" would mean, if it was not a unique smallest supremum of any subset of the set. |
| 2026-02-24 10:45:07 +0100 | <tomsmeding> | which... we knew 10 minutes ago already |
| 2026-02-24 10:44:57 +0100 | <tomsmeding> | so Bounded is not a "bounded lattice", it's a "partial order with minimal and maximal elements" |
| 2026-02-24 10:44:33 +0100 | <tomsmeding> | "lattice" implies that such upper bounds (and lower bounds) are unique |
| 2026-02-24 10:44:31 +0100 | <__monty__> | dminuoso: I think they're talking about lattices. |
| 2026-02-24 10:44:16 +0100 | <tomsmeding> | dminuoso: even if there is a maximal element of the entire set, that does not necessarily mean there is a _unique_ smallest upper bound of any subset of the set |
| 2026-02-24 10:43:48 +0100 | <dminuoso> | Leary: Are you saying that Bounded does not have uniqueness requirements? |
| 2026-02-24 10:43:09 +0100 | <tomsmeding> | fair! Even pairs need not have unique joins or meets |
| 2026-02-24 10:42:51 +0100 | Enrico63 | (~Enrico63@host-79-19-156-232.retail.telecomitalia.it) (Quit: Client closed) |
| 2026-02-24 10:42:40 +0100 | <Leary> | It's a bounded partial order with minimal and maximal elements---that doesn't imply unique suprema or infima. |
| 2026-02-24 10:42:21 +0100 | <tomsmeding> | which that picture is an example of |
| 2026-02-24 10:42:06 +0100 | <jreicher> | https://en.wikipedia.org/wiki/Hasse_diagram |
| 2026-02-24 10:41:51 +0100 | xff0x | (~xff0x@fsb6a9491c.tkyc517.ap.nuro.jp) (Ping timeout: 255 seconds) |
| 2026-02-24 10:41:39 +0100 | <jreicher> | dminuoso: hasse diagrams |
| 2026-02-24 10:41:19 +0100 | <tomsmeding> | because of this picture I always assumed https://commons.wikimedia.org/wiki/File:Pow3nonlattice.svg |
| 2026-02-24 10:40:55 +0100 | dminuoso | wonders why lattices are called lattices |
| 2026-02-24 10:40:38 +0100 | <tomsmeding> | thus all complete lattices are bounded, but not all bounded lattices are complete? Perhaps? |
| 2026-02-24 10:40:27 +0100 | <tomsmeding> | a _complete lattice_ additionally requires suprema and infima of larger sets |
| 2026-02-24 10:40:10 +0100 | <tomsmeding> | and thus finite sets |
| 2026-02-24 10:40:07 +0100 | <tomsmeding> | according to the wikipedia definition that I'm currently reading, a plain "lattice" requires joins and meets of _two_ elements |
| 2026-02-24 10:40:06 +0100 | <dminuoso> | I see. |
| 2026-02-24 10:40:04 +0100 | <dminuoso> | Ohh wait. |
| 2026-02-24 10:39:46 +0100 | <dminuoso> | tomsmeding: Aren't all lattices bounded by their suprema? |
| 2026-02-24 10:39:16 +0100 | <tomsmeding> | well, a bounded lattice |
| 2026-02-24 10:38:55 +0100 | <tomsmeding> | perhaps? |
| 2026-02-24 10:38:47 +0100 | <dminuoso> | tomsmeding: Okay so what you're saying is that Bounded is just synonmous for Lattice? |
| 2026-02-24 10:38:17 +0100 | <tomsmeding> | dminuoso: all subset of a given set S are partially ordered by inclusion, and they have a unique minimum ({}) and maximum (S), but they are not totally ordered |
| 2026-02-24 10:37:38 +0100 | <tomsmeding> | (but then, it's probably "minimum", which would not have helped dminuoso) |
