A334255 -id:A334255 - cp's OEIS frontend

A334254 Number of closure operators on a set of n elements which satisfy the T_1 separation axiom.

1, 2, 1, 8, 545, 702525, 66960965307

Offset: 0

Joshua Moerman, Apr 20 2020

The T_1 axiom states that all singleton sets {x} are closed.

For n>1, this property implies strictness (meaning that the empty set is closed).

			The a(3) = 8 set-systems of closed sets:
  {{1,2,3},{1},{2},{3},{}}
  {{1,2,3},{1,2},{1},{2},{3},{}}
  {{1,2,3},{1,3},{1},{2},{3},{}}
  {{1,2,3},{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},{}}

Dmitry I. Ignatov, On the Cryptomorphism between Davis' Subset Lattices, Atomic Lattices, and Closure Systems under T1 Separation Axiom, arXiv:2209.12256 [cs.DM], 2022.
Dmitry I. Ignatov, Supporting iPython code for counting closure systems w.r.t. the T_1 separation axiom, Github repository
Dmitry I. Ignatov, PDF of the supporting iPython notebook
S. Mapes, Finite atomic lattices and resolutions of monomial ideals, J. Algebra, 379 (2013), 259-276.
Eric Weisstein's World of Mathematics, Separation Axioms
Wikipedia, Separation Axiom

The number of all closure operators is given in A102896.
For T_0 closure operators, see A334252.
For strict T_1 closure operators, see A334255, the only difference is a(1).
Cf. A326960, A326961, A326979.

a(6) from Dmitry I. Ignatov, Jul 03 2022

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.

Original entry on oeis.org

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

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

A334253 Number of strict closure operators on a set of n elements which satisfy the T_0 separation axiom.

Original entry on oeis.org

1, 1, 3, 35, 2039, 1352390, 75945052607, 14087646108883940225

Offset: 0

Joshua Moerman, Apr 20 2020

The T_0 axiom states that the closure of {x} and {y} are different for distinct x and y.

A closure operator is strict if the empty set is closed.

			The a(0) = 1 through a(2) = 3 set-systems of closed sets:
{{}}  {{1},{}}  {{1,2},{1},{}}
                {{1,2},{2},{}}
                {{1,2},{1},{2},{}}

R. S. R. Myers, J. Adámek, S. Milius, and H. Urbat, Coalgebraic constructions of canonical nondeterministic automata, Theoretical Computer Science, 604 (2015), 81-101.
B. Venkateswarlu and U. M. Swamy, T_0-Closure Operators and Pre-Orders, Lobachevskii Journal of Mathematics, 39 (2018), 1446-1452.

The number of all strict closure operators is given in A102894.
For all T0 closure operators, see A334252.
For strict T1 closure operators, see A334255.
A strict closure operator which preserves unions is called topological, see A001035.
Cf. A326943, A326944, A326945.

a(n) = Sum_{k=0..n} Stirling1(n,k) * A102894(k). - Andrew Howroyd, Apr 20 2020

a(6)-a(7) from Andrew Howroyd, Apr 20 2020

A355517 Number of nonisomorphic systems enumerated by A334254; that is, the number of inequivalent closure operators on a set of n elements where all singletons are closed.

Original entry on oeis.org

1, 2, 1, 4, 50, 7443, 95239971

Offset: 0

Dmitry I. Ignatov, Jul 05 2022

The T_1 axiom states that all singleton sets {x} are closed.

For n>1, this property implies strictness (meaning that the empty set is closed).

			a(0) = 1 counts the empty set, while a(1) = 2 counts {{1}} and {{},{1}}.
For a(2) = 1 the closure system is as follows:  {{1,2},{1},{2},{}}.
The a(3) = 4 inequivalent set-systems of closed sets are:
  {{1,2,3},{1},{2},{3},{}}
  {{1,2,3},{1,2},{1},{2},{3},{}}
  {{1,2,3},{1,2},{1,3},{1},{2},{3},{}}
  {{1,2,3},{1,2},{1,3},{2,3},{1},{2},{3},{}}.

Dmitry I. Ignatov, Supporting iPython code for counting nonequivalent closure systems w.r.t. the T_1 separation axiom, Github repository
Eric Weisstein's World of Mathematics, Separation Axioms
Wikipedia, Separation Axiom

The number of all closure operators is given in A102896, while A193674 contains the number of all nonisomorphic ones.
For T_1 closure operators and their strict counterparts, see A334254 and A334255, respectively; the only difference is a(1).
Cf. A326960, A326961, A326979.

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

A334254 Number of closure operators on a set of n elements which satisfy the T_1 separation axiom.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Extensions

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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Extensions

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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Extensions

A334253 Number of strict closure operators on a set of n elements which satisfy the T_0 separation axiom.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A355517 Number of nonisomorphic systems enumerated by A334254; that is, the number of inequivalent closure operators on a set of n elements where all singletons are closed.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Extensions

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

Views

Author

Keywords