

A102896


Number of ACI algebras (or semilattices) on n generators with no annihilator.


29




OFFSET

0,2


COMMENTS

Or, number of Moore families on an nset, 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 setsystems on n vertices that are closed under union. The BIInumbers of these setsystems are given by A326875.  Gus Wiseman, Jul 31 2019


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). [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. 5380.


LINKS

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 coMoore Families.
P. Colomb, A. Irlande, O. Raynaud, Y. Renaud, Recursive decomposition tree of a Moore cofamily and closure algorithm, Annals of Mathematics and Artificial Intelligence, 2013, DOI 10.1007/s104720139362x.
N. Dershowitz, G. S. Huang and M. Harris, Enumeration Problems Related to Ground Horn Theories, arXiv:cs/0610054v2 [cs.LO], 20062008.
M. Habib and L. Nourine, The number of Moore families on n = 6, Discrete Math., 294 (2005), 291296.


FORMULA

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


EXAMPLE

From Gus Wiseman, Jul 31 2019: (Start)
The a(0) = 1 through a(2) = 7 setsystems closed under union:
{} {} {}
{{1}} {{1}}
{{2}}
{{1,2}}
{{1},{1,2}}
{{2},{1,2}}
{{1},{2},{1,2}}
(End)


MATHEMATICA

Table[Length[Select[Subsets[Subsets[Range[n], {1, n}]], SubsetQ[#, Union@@@Tuples[#, 2]]&]], {n, 0, 3}] (* Gus Wiseman, Jul 31 2019 *)


CROSSREFS

For setsystems closed under union:
 The covering case is A102894.
 The unlabeled case is A193674.
 The case also closed under intersection is A306445.
 Setsystems closed under overlapping union are A326866.
 The BIInumbers of these setsystems 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


KEYWORD

nonn,hard,more


AUTHOR

Mitch Harris, Jan 18 2005


EXTENSIONS

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


STATUS

approved



