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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. 2
 1, 1, 1, 4, 167 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS 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. REFERENCES 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). LINKS Victor Chepoi, Separation of Two Convex Sets in Convexity Structures Wikipedia, Closure operator EXAMPLE 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}} MATHEMATICA 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}] CROSSREFS Cf. A334255, A358152, A356544. Sequence in context: A185857 A347719 A221654 * A159011 A077257 A278124 Adjacent sequences: A358141 A358142 A358143 * A358145 A358146 A358147 KEYWORD nonn,hard,more AUTHOR Tian Vlasic, Oct 31 2022 STATUS approved

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Last modified March 22 20:34 EDT 2023. Contains 361433 sequences. (Running on oeis4.)