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A358152 Number of strict closure operators on a set of n elements such that every point and every set not containing that point can be separated by clopen sets. 4
1, 1, 2, 8, 121 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
A closure operator is strict if the empty set is closed.
A point p in X and a subset A of X not containing p are separated by a set H if p is an element of H and A is a subset of X\H.
Also the number of S_3 convexities on a set of n elements in the sense of Chepoi.
REFERENCES
G. M. Bergman, "Lattices, Closure Operators, and Galois Connections", pp. 173-212 in "An Invitation to General Algebra and Universal Constructions", Springer, (2015).
LINKS
EXAMPLE
The a(3) = 8 set-systems of closed sets:
{{}, {1, 2, 3}}
{{}, {1}, {2, 3}, {1, 2, 3}}
{{}, {2}, {1, 3},{1, 2, 3}}
{{}, {3}, {1, 2}, {1, 2, 3}}
{{}, {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[
Flatten[(x |-> ({x, #} & /@ Select[F, FreeQ[#, x] &])) /@ Range[n],
1], MemberQ[F,
_?(H |-> With[{H1 = Complement[Range[n], H]},
MemberQ[F, H1] && MemberQ[H, #[[1]]
] && SubsetQ[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
Sequence in context: A112094 A009658 A147794 * A027530 A228064 A271846
KEYWORD
nonn,hard,more,changed
AUTHOR
Tian Vlasic, Nov 01 2022
STATUS
approved

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Last modified May 20 11:55 EDT 2024. Contains 372712 sequences. (Running on oeis4.)