A356544 Number of strict closure operators on a set of n elements such that all pairs of nonempty disjoint closed sets can be separated by clopen sets.

0, 1, 4, 35, 857, 84230, 70711467

Offset: 0

Author

Tian Vlasic, Aug 11 2022

Keywords

• nonn
• hard
• more

Comments

A closure operator is strict if the empty set is closed.

Two nonempty disjoint subsets A and B of X are separated by a set H if A is a subset of H and B is a subset of X\H.

Also the number of S_4 (Kakutani separation property) convexities on a set of n elements in the sense of Chepoi.

Examples

			The a(3) = 35 set-systems of closed sets:
{{}, {1, 2, 3}}
{{}, {1}, {1, 2, 3}}
{{}, {2}, {1, 2, 3}}
{{}, {3}, {1, 2, 3}}
{{}, {1, 2}, {1, 2, 3}}
{{}, {1, 3}, {1, 2, 3}}
{{}, {2, 3}, {1, 2, 3}}
{{}, {1}, {1, 2}, {1, 2, 3}}
{{}, {1}, {1, 3}, {1, 2, 3}}
{{}, {1}, {2, 3}, {1, 2, 3}}
{{}, {2}, {1, 2}, {1, 2, 3}}
{{}, {2}, {1, 3}, {1, 2, 3}}
{{}, {2}, {2, 3}, {1, 2, 3}}
{{}, {3}, {1, 2}, {1, 2, 3}}
{{}, {3}, {1, 3}, {1, 2, 3}}
{{}, {3}, {2, 3}, {1, 2, 3}}
{{}, {1}, {2}, {1, 3}, {1, 2, 3}}
{{}, {1}, {2}, {2, 3}, {1, 2, 3}}
{{}, {1}, {3}, {1, 2}, {1, 2, 3}}
{{}, {1}, {3}, {2, 3}, {1, 2, 3}}
{{}, {1}, {1, 2}, {1, 3}, {1, 2, 3}}
{{}, {2}, {3}, {1, 2}, {1, 2, 3}}
{{}, {2}, {3}, {1, 3}, {1, 2, 3}}
{{}, {2}, {1, 2}, {2, 3}, {1, 2, 3}}
{{}, {3}, {1, 3}, {2, 3}, {1, 2, 3}}
{{}, {1}, {2}, {1, 2}, {1, 3}, {1, 2, 3}}
{{}, {1}, {2}, {1, 2}, {2, 3}, {1, 2, 3}}
{{}, {1}, {3}, {1, 2}, {1, 3}, {1, 2, 3}}
{{}, {1}, {3}, {1, 3}, {2, 3}, {1, 2, 3}}
{{}, {2}, {3}, {1, 2}, {2, 3}, {1, 2, 3}}
{{}, {2}, {3}, {1, 3}, {2, 3}, {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}}
		

References

• G. M. Bergman. Lattices, Closure Operators, and Galois Connections. Springer, Cham. 2015. 173-212.

Links

• Victor Chepoi, Separation of Two Convex Sets in Convexity Structures
• Wikipedia, Closure operator

Crossrefs

Cf. A334255, A358144, A358152.

Programs

Mathematica

SeparatedPairQ[A_][B_] := AnyTrue[A, And @@ MapThread[SubsetQ, {#, B}] &];
Table[Length[With[{X = Range[n]},
Select[Cases[Subsets@Subsets@X, {{}, _, X}],
   F |-> SubsetQ[F, Intersection @@@ Subsets[F, {2}]]
&& AllTrue[Select[Subsets[Drop[F, 1], {2}], Apply[DisjointQ]], SeparatedPairQ[Select[{#, Complement[X, #]} & /@ F, MemberQ[F, #[[2]]] &]]]]]], {n, 0, 4}]

Extensions

a(5)-a(6) from Christian Sievers, Jun 13 2024