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A102895 Number of ACI algebras or semilattices on n generators with no identity element. 7
2, 2, 8, 90, 4542, 2747402, 151930948472, 28175295407840207894 (list; graph; refs; listen; history; text; internal format)



An ACI algebra or semilattice is a system with a single binary, idempotent, commutative and associative operation.

Or, number of families of subsets of {1, ..., n} that are closed under intersection and contain the empty set.


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)

M. Habib and L. Nourine, The number of Moore families on n = 6, Discrete Math., 294 (2005), 291-296.

E. H. Moore, Introduction to a Form of General Analysis, AMS Colloquium Publication 2 (1910), pp. 53-80.


Table of n, a(n) for n=0..7.

N. Dershowitz, G. S. Huang and M. Harris, Enumeration Problems Related to Ground Horn Theories, arXiv:cs/0610054v2 [cs.LO].


For asymptotics see A102897.

a(n) = 2*A102894(n).


a(2) = 8: Let the points be labeled a, b and let 0 denote the empty set. We want the number of collections of subsets of {a, b} which are closed under intersection and contain the empty subset. 0 subsets: 0 ways, 1 subset: 1 way (0), 2 subsets: 3 ways (0,a; 0,b; 0,ab), 3 subsets: 3 ways (0,a,b; 0,a,ab; 0,b,ab), 4 subsets: 1 way (0,a,b,ab), for a total of 8.


Cf. A102894, A102896, A102897, A108798, A193674, A108800, A193675.

Sequence in context: A295382 A123642 A007848 * A303060 A270555 A270405

Adjacent sequences:  A102892 A102893 A102894 * A102896 A102897 A102898




Mitch Harris, Jan 18 2005


Additional comments from Don Knuth, Jul 01 2005



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Last modified August 21 02:32 EDT 2018. Contains 313930 sequences. (Running on oeis4.)