%I
%S 1,2,7,61,2480,1385552,75973751474,14087648235707352472
%N Number of ACI algebras (or semilattices) on n generators with no annihilator.
%C 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.
%C Or, number of closure operators on a set of n elements.
%C An ACI algebra or semilattice is a system with a single binary, idempotent, commutative and associative operation.
%C 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
%D G. Birkhoff, Lattice Theory. American Mathematical Society, Colloquium Publications, Vol. 25, 3rd ed., Providence, RI, 1967.
%D Maria Paola Bonacina and Nachum Dershowitz, Canonical Inference for Implicational Systems, in Automated Reasoning, Lecture Notes in Computer Science, Volume 5195/2008, SpringerVerlag.
%D 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]
%D E. H. Moore, Introduction to a Form of General Analysis, AMS Colloquium Publication 2 (1910), pp. 5380.
%H Pierre Colomb, Alexis Irlande, Olivier Raynaud and Yoan Renaud, <a href="http://www.colomb.me/pierre/data/paper/icfca2011.pdf">About the Recursive Decomposition of the lattice of coMoore Families</a>.
%H P. Colomb, A. Irlande, O. Raynaud, Y. Renaud, <a href="https://doi.org/10.1007/s104720139362x">Recursive decomposition tree of a Moore cofamily and closure algorithm</a>, Annals of Mathematics and Artificial Intelligence, 2013, DOI 10.1007/s104720139362x.
%H N. Dershowitz, G. S. Huang and M. Harris, <a href="http://arxiv.org/abs/cs/0610054">Enumeration Problems Related to Ground Horn Theories</a>, arXiv:cs/0610054v2 [cs.LO], 20062008.
%H M. Habib and L. Nourine, <a href="https://doi.org/10.1016/j.disc.2004.11.010">The number of Moore families on n = 6</a>, Discrete Math., 294 (2005), 291296.
%F a(n) = Sum_{k=0..n} C(n, k)*A102894(k), where C(n, k) is the binomial coefficient.
%F For asymptotics see A102897.
%F a(n) = A102897(n)/2.  _Gus Wiseman_, Jul 31 2019
%e From _Gus Wiseman_, Jul 31 2019: (Start)
%e The a(0) = 1 through a(2) = 7 setsystems closed under union:
%e {} {} {}
%e {{1}} {{1}}
%e {{2}}
%e {{1,2}}
%e {{1},{1,2}}
%e {{2},{1,2}}
%e {{1},{2},{1,2}}
%e (End)
%t Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],SubsetQ[#,Union@@@Tuples[#,2]]&]],{n,0,3}] (* _Gus Wiseman_, Jul 31 2019 *)
%Y For setsystems closed under union:
%Y  The covering case is A102894.
%Y  The unlabeled case is A193674.
%Y  The case also closed under intersection is A306445.
%Y  Setsystems closed under overlapping union are A326866.
%Y  The BIInumbers of these setsystems are given by A326875.
%Y Cf. A102895, A102897, A108798, A108800, A193675, A000798, A014466, A326878, A326880, A326881.
%K nonn,hard,more
%O 0,2
%A _Mitch Harris_, Jan 18 2005
%E _N. J. A. Sloane_ added a(6) from the Habib et al. reference, May 26 2005
%E Additional comments from _Don Knuth_, Jul 01 2005
%E a(7) from Pierre Colomb (pierre(AT)colomb.me), Sep 04 2010
