A358152 Number of strict closure operators on a set of n elements such that every point and every closed set not containing that point can be separated by clopen sets.

1, 1, 2, 8, 121, 18460, 159273237

Offset: 0

Author

Tian Vlasic, Nov 01 2022

Keywords

• nonn
• hard
• more

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.

Examples

			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}}
		

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

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

Crossrefs

Cf. A334255, A358144, A356544.

Programs

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}]

Extensions

a(5)-a(6) from Christian Sievers, Jul 20 2024