Post #2278879
2026-05-06 15:21 UTC
@lisyarus@mastodon.gamedev.place Sounds plausible... I would like to extract the set of elements, I'm thinking that if you delete the bottom element of the lattice then the set of elements is given up to bijection by the set of remaining minimal elements, which are the singleton subsets
Replies (2)
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@lisyarus@mastodon.gamedev.place 2026-05-06 15:22
@julesh@mathstodon.xyz Hmmm, the maximal element is already the set of elements, isn't it?
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@mevenlennonbertrand@lipn.info 2026-05-06 21:02
@julesh@mathstodon.xyz @lisyarus@mastodon.gamedev.place This is a baby version of Stone duality, ie the fact that Set^op is equivalent to the category of complete atomic boolean algebras : in one direction, you map a set to its powerset, which is a CABA ; in the other direction, you map a CABA to its set of atoms (elements x st y < x implies y = ⊥). Indeed, the atoms in a powerset are exactly the singletons, and a set is isomorphic to the set of its singleton. (The good thing is that this is nicely structural, and does not need to play set-theoretic trickery with unions or some such.)