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

Author

Joshua Moerman, Apr 20 2020

Keywords

• nonn
• more
• hard

Comments

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).

Examples

			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},{}}
		

Links

• 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

Crossrefs

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.

Extensions

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