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| 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) |
| 2026-02-24 10:37:27 +0100 | <tomsmeding> | perhaps "lower bound" is not the technically correct term here |
| 2026-02-24 10:37:15 +0100 | <tomsmeding> | sure, but even those need not have a total ordering |
| 2026-02-24 10:36:44 +0100 | <__monty__> | It's restricted to partial orders that have a single lower and upper bound. |
| 2026-02-24 10:36:43 +0100 | <tomsmeding> | dminuoso: yes, a lower bound :) https://en.wikipedia.org/wiki/Lattice_(order) |
| 2026-02-24 10:36:32 +0100 | <__monty__> | tomsmeding: But Bounded doesn't deal with just any partial order. |
| 2026-02-24 10:36:30 +0100 | <dminuoso> | Im not even sure what that would mean. |
| 2026-02-24 10:36:22 +0100 | <dminuoso> | tomsmeding: Is there a notion of `minimum` in a partially ordered set, then? |
| 2026-02-24 10:36:09 +0100 | sp1ff | (~user@c-24-20-218-28.hsd1.wa.comcast.net) sp1ff |
| 2026-02-24 10:35:55 +0100 | <humasect> | how to learn ? |
| 2026-02-24 10:35:54 +0100 | <tomsmeding> | partial orders are a thing |
| 2026-02-24 10:35:54 +0100 | <dminuoso> | Oh. |
| 2026-02-24 10:35:50 +0100 | <tomsmeding> | dminuoso: not every ordering is total |
| 2026-02-24 10:35:48 +0100 | <humasect> | perhaps look into its meaning, disassemble it to understanding |
| 2026-02-24 10:35:43 +0100 | <dminuoso> | It reads nonsensical to me. |
| 2026-02-24 10:35:32 +0100 | <dminuoso> | humasect: Yes, and its not much of an explanation. |
| 2026-02-24 10:35:24 +0100 | tomsmeding | should have thought of that |
| 2026-02-24 10:35:05 +0100 | <tomsmeding> | lol, dminuoso ^ |
| 2026-02-24 10:34:47 +0100 | <humasect> | it says right in the first sentence https://hackage-content.haskell.org/package/base-4.22.0.0/docs/Data-Bounded.html |
| 2026-02-24 10:33:53 +0100 | <tomsmeding> | but then you could argue that you should give that Ord instance anyway |
| 2026-02-24 10:33:40 +0100 | <tomsmeding> | perhaps it makes sense that an Enum has a first and a last value for enumeration, but while it has an ordering compatible with that Enum instance, that ordering makes no semantical sense and is hence not defined? |
