A367422 Number of inequivalent strict interval closure operators on a set of n elements.

1, 1, 3, 14, 146, 6311, 2302155

Offset: 0

Author

Dmitry I. Ignatov, Nov 18 2023

Keywords

• nonn
• hard
• more

Comments

A closure operator cl is strict if {} is closed, i.e., cl({})={}; it is interval closure operator if for every set S, the statement that for all x,y in S, cl({x,y}) is a subset of S implies that S is closed.

a(n) is also the number of interval convexities on a set of n elements (see Chepoi).

Examples

			The a(2) = 3 set-systems include {}{12}, {}{1}{2}{12}, {}{1}{12} (equivalent to {}{2}{12}).
The a(3) = 14 set-systems are the following (system {{}, {1,2,3}}, and sets {} and {1,2,3} are omitted).
    {1}
    {1}{12}
    {12}
    {1}{12}{13}
    {1}{2}
    {1}{2}{12}
    {1}{2}{3}{12}
    {1}{2}{3}
    {1}{2}{13}
    {1}{2}{3}{13}{23}
    {1}{2}{12}{23}
    {1}{23}
    {1}{2}{3}{12}{13}{23}.
		

References

• B. Ganter and R. Wille, Formal Concept Analysis - Mathematical Foundations, Springer, 1999, pages 1-15.

Links

• Victor Chepoi, Separation of Two Convex Sets in Convexity Structures
• Dmitry I. Ignatov, Supporting iPython code for counting (inequivalent) strict interval closure operators up to n=6, Github repository.
• Wikipedia, Closure operator

Crossrefs

Cf. A364656 (all strict interval closure families), A334255, A358144, A358152, A356544.