OFFSET
0,4
COMMENTS
A closure operator is strict if the empty set is closed.
Two distinct points x,y in X are separated by a set H if x is an element of H and y is not an element of H.
Also the number of S_2 convexities on a set of n elements in the sense of Chepoi.
REFERENCES
G. M. Bergman. Lattices, Closure Operators, and Galois Connections. Springer, Cham. 2015. 173-212 in "An Invitation to General Algebra and Universal Constructions", Springer, (2015).
LINKS
Victor Chepoi, Separation of Two Convex Sets in Convexity Structures
Wikipedia, Closure operator
EXAMPLE
The a(3) = 4 set-systems of closed sets:
{{}, {1}, {2}, {3}, {1, 2}, {1, 3}, {1, 2, 3}}
{{}, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 2, 3}}
{{}, {1}, {2}, {3}, {1, 3}, {2, 3}, {1, 2, 3}}
{{}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}
MATHEMATICA
SeparatedPairQ[F_, n_] := AllTrue[
Subsets[Range[n], {2}],
MemberQ[F,
_?(H |-> With[{H1 = Complement[Range[n], H]},
MemberQ[F, H1] && MemberQ[H, #[[1]]
] && MemberQ[H1, #[[2]]
]])] &];
Table[Length@Select[Select[
Subsets[Subsets[Range[n]]],
And[
MemberQ[#, {}],
MemberQ[#, Range[n]],
SubsetQ[#, Intersection @@@ Tuples[#, 2]]] &
], SeparatedPairQ[#, n] &] , {n, 0, 4}]
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Tian Vlasic, Oct 31 2022
EXTENSIONS
a(5) from Christian Sievers, Feb 04 2024
a(6) from Christian Sievers, Jun 13 2024
STATUS
approved