A334254 -id:A334254 - cp's OEIS frontend

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.

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.

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

Original entry on oeis.org

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

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

A235604 Number of equivalence classes of lattices of subsets of the power set 2^[n].

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Jan 21 2014

This is also the number of inequivalent atomic lattices on n atoms or inequivalent strict closure systems under T1 separation axiom on n elements. - Dmitry I. Ignatov, Sep 27 2022

Donald M. Davis, Enumerating lattices of subsets, arXiv preprint arXiv:1311.6664 [math.CO], 2013.
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.

The number of inequivalent closure operators on a set of n elements where all singletons are closed is given in A355517.
The number of all strict closure operators is given in A102894.
For T_1 closure operators, see A334254.

a(5) from Andrew Weimholt, Jan 27 2014

a(6) from Dmitry I. Ignatov, Sep 27 2022

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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.

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

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A235604 Number of equivalence classes of lattices of subsets of the power set 2^[n].

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