A358144 -id:A358144 - cp's OEIS frontend

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

Tian Vlasic, Aug 11 2022

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.

			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}}

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

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

Cf. A334255, A358144, A358152.

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}]

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

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.

Original entry on oeis.org

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

Offset: 0

Tian Vlasic, Nov 01 2022

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.

			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}}

G. M. Bergman, "Lattices, Closure Operators, and Galois Connections", pp. 173-212 in "An Invitation to General Algebra and Universal Constructions", Springer, (2015).

Victor Chepoi, Separation of Two Convex Sets in Convexity Structures

Cf. A334255, A358144, A356544.

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}]

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

A364656 Number of strict interval closure operators on a set of n elements.

Original entry on oeis.org

1, 1, 4, 45, 2062, 589602, 1553173541

Offset: 0

Tian Vlasic, Jul 31 2023

A closure operator cl on a set X is strict if the empty set is closed; it is an interval if for every subset S of X, 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).

			The a(3) = 45 set-systems are the following ({} and {1,2,3} not shown).
    {1}   {1}{2}   {1}{2}{3}    {1}{2}{3}{12}   {1}{2}{3}{12}{13}
    {2}   {1}{3}   {1}{2}{12}   {1}{2}{3}{13}   {1}{2}{3}{12}{23}
    {3}   {2}{3}   {1}{2}{13}   {1}{2}{3}{23}   {1}{2}{3}{13}{23}
    {12}  {1}{12}  {1}{2}{23}   {1}{2}{12}{13}
    {13}  {1}{13}  {1}{3}{12}   {1}{2}{12}{23}
    {23}  {1}{23}  {1}{3}{13}   {1}{3}{12}{13}        {1}{2}{3}{12}{13}{23}
          {2}{12}  {1}{3}{23}   {1}{3}{13}{23}
          {2}{13}  {2}{3}{12}   {2}{3}{12}{23}
          {2}{23}  {2}{3}{13}   {2}{3}{13}{23}
          {3}{12}  {2}{3}{23}
          {3}{13}  {1}{12}{13}
          {3}{23}  {2}{12}{23}
                   {3}{13}{23}

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

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

Cf. A334255, A358144, A358152, A356544.

Table[With[{closure = {X, set} |->
      Intersection @@ Select[X, SubsetQ[#, set] &]},
   Select[
    Select[
     Join[{{}, Range@n}, #] & /@ Subsets@Subsets[Range@n, {1, n - 1}],
      SubsetQ[#, Intersection @@@ Subsets[#, {2}]] &],
    X |->
     AllTrue[Complement[Subsets@Range@n, X],
      S |-> \[Not]
        AllTrue[Subsets[S, {1, 2}], SubsetQ[S, closure[X, #]] &]]]] //
   Length, {n, 4}]

New offset and a(5)-a(6) from Dmitry I. Ignatov, Nov 14 2023

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

Original entry on oeis.org

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

Offset: 0

Dmitry I. Ignatov, Nov 18 2023

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).

			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}.

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

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

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

cp's OEIS Frontend

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.

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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.

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A364656 Number of strict interval closure operators on a set of n elements.

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

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