0,1

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 intersectionand contain the empty set.

G. Birkhoff, Lattice Theory. American Mathematical Society, Colloquium Publications, Vol. 25, 3rd ed., Providence, RI, 1967.

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.

Maria Paola Bonacina and Nachum Dershowitz, Canonical Inference for Implicational Systems, in Automated Reasoning, Lecture Notes in Computer Science, Volume 5195/2008, Springer-Verlag.

N. Dershowitz, G. S. Huang and M. Harris, Draft.

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, A108799, A108800, A108801.

Sequence in context: A012410 A123642 A007848 this_sequence A047692 A069561 A011148

Adjacent sequences: A102892 A102893 A102894 this_sequence A102896 A102897 A102898

nonn,hard,more

Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu), Jan 18 2005

Additional comments from D. E. Knuth, Jul 01, 2005