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A102896
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Number of ACI algebras (or semilattices) on n generators with no annihilator.
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43
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OFFSET
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0,2
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COMMENTS
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Or, number of Moore families on an n-set, that is, families of subsets that contain the universal set {1,...,n} and are closed under intersection.
Or, number of closure operators on a set of n elements.
An ACI algebra or semilattice is a system with a single binary, idempotent, commutative and associative operation.
Also the number of set-systems on n vertices that are closed under union. The BII-numbers of these set-systems are given by A326875. - Gus Wiseman, Jul 31 2019
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REFERENCES
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G. Birkhoff, Lattice Theory. American Mathematical Society, Colloquium Publications, Vol. 25, 3rd ed., Providence, RI, 1967.
Maria Paola Bonacina and Nachum Dershowitz, Canonical Inference for Implicational Systems, in Automated Reasoning, Lecture Notes in Computer Science, Volume 5195/2008, Springer-Verlag.
P. Colomb, A. Irlande and O. Raynaud, Counting of Moore Families for n=7, International Conference on Formal Concept Analysis (2010). [From Pierre Colomb (pierre(AT)colomb.me), Sep 04 2010]
E. H. Moore, Introduction to a Form of General Analysis, AMS Colloquium Publication 2 (1910), pp. 53-80.
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LINKS
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Andrew J. Blumberg, Michael A. Hill, Kyle Ormsby, Angélica M. Osorno, and Constanze Roitzheim, Homotopical Combinatorics, Notices Amer. Math. Soc. (2024) Vol. 71, No. 2, 260-266. See p. 261.
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FORMULA
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a(n) = Sum_{k=0..n} C(n, k)*A102894(k), where C(n, k) is the binomial coefficient.
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EXAMPLE
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The a(0) = 1 through a(2) = 7 set-systems closed under union:
{} {} {}
{{1}} {{1}}
{{2}}
{{1,2}}
{{1},{1,2}}
{{2},{1,2}}
{{1},{2},{1,2}}
(End)
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MATHEMATICA
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Table[Length[Select[Subsets[Subsets[Range[n], {1, n}]], SubsetQ[#, Union@@@Tuples[#, 2]]&]], {n, 0, 3}] (* Gus Wiseman, Jul 31 2019 *)
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CROSSREFS
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For set-systems closed under union:
- The case also closed under intersection is A306445.
- Set-systems closed under overlapping union are A326866.
- The BII-numbers of these set-systems are given by A326875.
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KEYWORD
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nonn,hard,more,changed
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AUTHOR
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EXTENSIONS
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Additional comments from Don Knuth, Jul 01 2005
a(7) from Pierre Colomb (pierre(AT)colomb.me), Sep 04 2010
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STATUS
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approved
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