A358152 -id:A358152 - 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

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.

Original entry on oeis.org

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

Offset: 0

Tian Vlasic, Oct 31 2022

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.

			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}}

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).

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

Cf. A334255, A358152, A356544.

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}]

a(5) from Christian Sievers, Feb 04 2024

a(6) from Christian Sievers, Jun 13 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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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.

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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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