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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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29
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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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Table of n, a(n) for n=0..7.
Pierre Colomb, Alexis Irlande, Olivier Raynaud and Yoan Renaud, About the Recursive Decomposition of the lattice of co-Moore Families.
P. Colomb, A. Irlande, O. Raynaud, Y. Renaud, Recursive decomposition tree of a Moore co-family and closure algorithm, Annals of Mathematics and Artificial Intelligence, 2013, DOI 10.1007/s10472-013-9362-x.
N. Dershowitz, G. S. Huang and M. Harris, Enumeration Problems Related to Ground Horn Theories, arXiv:cs/0610054v2 [cs.LO], 2006-2008.
M. Habib and L. Nourine, The number of Moore families on n = 6, Discrete Math., 294 (2005), 291-296.
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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.
For asymptotics see A102897.
a(n) = A102897(n)/2. - Gus Wiseman, Jul 31 2019
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EXAMPLE
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From Gus Wiseman, Jul 31 2019: (Start)
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 covering case is A102894.
- The unlabeled case is A193674.
- 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.
Cf. A102895, A102897, A108798, A108800, A193675, A000798, A014466, A326878, A326880, A326881.
Sequence in context: A046846 A111010 A089307 * A088107 A132524 A153694
Adjacent sequences: A102893 A102894 A102895 * A102897 A102898 A102899
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KEYWORD
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nonn,hard,more
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AUTHOR
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Mitch Harris, Jan 18 2005
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EXTENSIONS
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N. J. A. Sloane added a(6) from the Habib et al. reference, May 26 2005
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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