edit: neglected to mention that choosing which lattice (and or dual) to use only really matters when considering products/sums of lattices to form new lattices On Mon, Dec 21, 2020 at 11:12 AM Carter Schonwald < carter.schonwald@gmail.com> wrote:
why are we equating the lattice operators for True and false with the lattice operators for set? (for both structures, we have the dual partial order is also a lattice, so unless we have ) (i'm going to get the names of these equations wrong, but )
the "identity" law is going to be max `intersect` y == y , min `union` y === y
the "absorbing" law is going to be min `intersect` y == min , max `union` y == max
these rules work the same for (min = emptyset, max == full set, union == set union, intersect == set intersecct) OR for its dual lattice (min == full set, max == emtpy set, union = set intersection, intersect == set union)
at some level arguing about the empty list case turns into artifacts of "simple" definitions
that said, i suppose a case could be made that for intersect :: [a] -> a , that as the list argument gets larger the result should be getting *smaller*, so list intersect of lattice elements should be "anti-monotone", and list union should be monotone (the result gets bigger). I dont usually see tht
either way, I do strongly feel that either way, arguing by how we choose to relate the boolean lattice and seet lattices is perhaps the wrong choice... because both lattices are equivalent to theeir dual lattice
On Mon, Dec 21, 2020 at 5:12 AM Tom Ellis < tom-lists-haskell-cafe-2017@jaguarpaw.co.uk> wrote:
Am 06.12.20 um 19:58 schrieb Sven Panne:
To me it's just the other way around: It violates aesthetics if it doesn't follow the mathematical definition in all cases, which is why I don't
On Sun, Dec 20, 2020 at 11:05:39PM +0100, Ben Franksen wrote: like
NonEmpty here.
I think you've got that wrong.
x `elem` intersections [] = {definition} forall xs in []. x `elem` xs = {vacuous forall} true
Any proposition P(x) is true for all x in []. So the mathematical definition of intersections::[Set a]-> Set a would not be the empty set but the set of all x:a, which in general we have no way to construct.
Yes, and to bring this back to Sven's original claim
| Why NonEmpty? I would expect "intersections [] = Set.empty", because | the result contains all the elements which are in all sets, | i.e. none. That's at least my intuition, is there some law which | this would violate?
the correct definition of "intersections []" should be "all elements which are in all of no sets" i.e. _every_ value of the given type!
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