

A102895


Number of ACI algebras or semilattices on n generators with no identity element.


7




OFFSET

0,1


COMMENTS

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.


REFERENCES

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, SpringerVerlag.
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), 291296.
E. H. Moore, Introduction to a Form of General Analysis, AMS Colloquium Publication 2 (1910), pp. 5380.


LINKS

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].


FORMULA

For asymptotics see A102897.
a(n) = 2*A102894(n).


EXAMPLE

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.


CROSSREFS

Cf. A102894, A102896, A102897, A108798, A193674, A108800, A193675.
Sequence in context: A295382 A123642 A007848 * A270555 A270405 A047692
Adjacent sequences: A102892 A102893 A102894 * A102896 A102897 A102898


KEYWORD

nonn,hard,more


AUTHOR

Mitch Harris, Jan 18 2005


EXTENSIONS

Additional comments from Don Knuth, Jul 01 2005


STATUS

approved



