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A102894
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Number of ACI algebras or semilattices on n generators, with no identity or annihilator.
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22
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OFFSET
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0,3
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COMMENTS
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Or, number of families of subsets of {1, ..., n} that are closed under intersection and contain both the universe and the empty set.
An ACI algebra or semilattice is a system with a single binary, idempotent, commutative and associative operation.
Also the number of set-systems covering n vertices that are closed under union. The BII-numbers of these set-systems are given by A326875. - Gus Wiseman, Aug 01 2019
Number of strict closure operators on a set of n elements, where the closure operator is said to be strict if the empty set is closed. - Tian Vlasic, Jul 30 2022
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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.
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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FORMULA
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Inverse binomial transform of A102896.
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EXAMPLE
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The a(3) = 45 set-systems with {} and {1,2,3} that are closed under intersection are the following ({} and {1,2,3} not shown). The BII-numbers of these set-systems are given by A326880.
0 {1} {1}{2} {1}{2}{3} {1}{2}{3}{12} {1}{2}{3}{12}{13}
{2} {1}{3} {1}{2}{12} {1}{2}{3}{13} {1}{2}{3}{12}{23}
{3} {2}{3} {1}{2}{13} {1}{2}{3}{23} {1}{2}{3}{13}{23}
{12} {1}{12} {1}{2}{23} {1}{2}{12}{13}
{13} {1}{13} {1}{3}{12} {1}{2}{12}{23}
{23} {1}{23} {1}{3}{13} {1}{3}{12}{13} {1}{2}{3}{12}{13}{23}
{2}{12} {1}{3}{23} {1}{3}{13}{23}
{2}{13} {2}{3}{12} {2}{3}{12}{23}
{2}{23} {2}{3}{13} {2}{3}{13}{23}
{3}{12} {2}{3}{23}
{3}{13} {1}{12}{13}
{3}{23} {2}{12}{23}
{3}{13}{23}
(End)
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MATHEMATICA
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Table[Length[Select[Subsets[Subsets[Range[n], {1, n}]], Union@@#==Range[n]&&SubsetQ[#, Union@@@Tuples[#, 2]]&]], {n, 0, 3}] (* Gus Wiseman, Aug 01 2019 *)
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CROSSREFS
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Regarding set-systems covering n vertices closed under union:
- The non-covering case is A102896.
- The BII-numbers of these set-systems are A326875.
- The case with intersection instead of union is A326881.
Cf. A003465, A072447, A102895, A102897, A108800, A193674, A193675, A306445, A326870, A326880, A326883.
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KEYWORD
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nonn,hard,more
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AUTHOR
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EXTENSIONS
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Additional comments from Don Knuth, Jul 01 2005
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STATUS
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approved
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