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A334255 Number of strict closure operators on a set of n elements which satisfy the T_1 separation axiom.

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

Offset: 0

Joshua Moerman, Apr 24 2020

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

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

The number of all strict closure operators is given in A102894.
For all strict T_0 closure operators, see A334253.
For T_1 closure operators, see A334254.
Cf. A326960, A326961, A326979.

Table[Length[
  Select[Subsets[Subsets[Range[n]]],
   And[MemberQ[#, {}], MemberQ[#, Range[n]],
     SubsetQ[#, Intersection @@@ Tuples[#, 2]],
     SubsetQ[#, Map[{#} &, Range[n]]]] &]], {n, 0, 4}] (* Tian Vlasic, Jul 29 2022 *)

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

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

Original entry on oeis.org

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

A334252 Number of closure operators on a set of n elements which satisfy the T_0 separation axiom.

Original entry on oeis.org

1, 2, 5, 44, 2179, 1362585, 75953166947, 14087646640499308474

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.

			The a(0) = 1 through a(2) = 5 set-systems of closed sets:
{{}}  {{}}      {{1,2},{1}}
      {{1},{}}  {{1,2},{2}}
                {{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 closure operators is given in A102896.
For strict T0 closure operators, see A334253.
For T1 closure operators, see A334254.
Cf. A326943, A326944, A326945.

a(n) = Sum_{k=0..n} Stirling1(n,k) * A102896(k). - Andrew Howroyd, Apr 20 2020

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

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

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A334255 Number of strict closure operators on a set of n elements which satisfy the T_1 separation axiom.

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

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A334252 Number of closure operators on a set of n elements which satisfy the T_0 separation axiom.

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

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Extensions