A396537 Number of inequivalent 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, 2, 18, 1767

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 non-isomorphic 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

Table of n, a(n) for n=0..5.

Victor Chepoi, Separation of Two Convex Sets in Convexity Structures

Tian Vlasic, Python program

Wikipedia, Closure operator

Example

a(3)=2 as there are 2 distinct S_2 convexities on a set of three elements. If we set X={a,b,c}, then the two convexities could be e.g.,
  {{}, {1}, {2}, {3}, {1, 2}, {1, 3}, {1, 2, 3}} and
  {{}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}.

Crossrefs

Cf. A358144, A334255, A358152, A356544.

Sequence in context: A334623 A260610 A333164 * A076954 A206847 A259654

Adjacent sequences: A396534 A396535 A396536 * A396538 A396539 A396540

Keyword

nonn,more

Author

Tian Vlasic, May 28 2026

Status

approved