A358144 Number of strict closure operators on a set of n elements such that all pairs of distinct points can be separated by clopen sets.

1, 1, 1, 4, 167, 165791, 19194240969

Offset: 0

Author

Tian Vlasic, Oct 31 2022

Keywords

• nonn
• hard
• more

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.

Examples

			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}}
		

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

Crossrefs

Cf. A334255, A358152, A356544.

Programs

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}]

Extensions

a(5) from Christian Sievers, Feb 04 2024

a(6) from Christian Sievers, Jun 13 2024