0,3

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

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

Also the number of set-systems covering n vertices that are closed under union. The BII-numbers of these set-systems are given by A326875. - Gus Wiseman, Aug 01 2019

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)

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.

Maria Paola Bonacina, Nachum Dershowitz, Canonical ground Horn theories, Lecture Notes in Computer Science 7797, 35-71 (2013).

N. Dershowitz, G. S. Huang and M. Harris, Enumeration Problems Related to Ground Horn Theories, arXiv:cs/0610054v2 [cs.LO], 2006-2008.

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

Inverse binomial transform of A102896.

For asymptotics see A102897.

From Gus Wiseman, Aug 01 2019: (Start)

The a(3) = 45 set-systems with {} and {1,2,3} that are closed under intersection are the following ({} and {1,2,3} not shown). The BII-numbers of these set-systems are given by A326880.

0   {1}   {1}{2}   {1}{2}{3}    {1}{2}{3}{12}   {1}{2}{3}{12}{13}

    {2}   {1}{3}   {1}{2}{12}   {1}{2}{3}{13}   {1}{2}{3}{12}{23}

    {3}   {2}{3}   {1}{2}{13}   {1}{2}{3}{23}   {1}{2}{3}{13}{23}

    {12}  {1}{12}  {1}{2}{23}   {1}{2}{12}{13}

    {13}  {1}{13}  {1}{3}{12}   {1}{2}{12}{23}

    {23}  {1}{23}  {1}{3}{13}   {1}{3}{12}{13}        {1}{2}{3}{12}{13}{23}

          {2}{12}  {1}{3}{23}   {1}{3}{13}{23}

          {2}{13}  {2}{3}{12}   {2}{3}{12}{23}

          {2}{23}  {2}{3}{13}   {2}{3}{13}{23}

          {3}{12}  {2}{3}{23}

          {3}{13}  {1}{12}{13}

          {3}{23}  {2}{12}{23}

                   {3}{13}{23}

(End)

Table[Length[Select[Subsets[Subsets[Range[n], {1, n}]], Union@@#==Range[n]&&SubsetQ[#, Union@@@Tuples[#, 2]]&]], {n, 0, 3}] (* Gus Wiseman, Aug 01 2019 *)

Regarding set-systems covering n vertices closed under union:

- The non-covering case is A102896.

- The BII-numbers of these set-systems are A326875.

- The case with intersection instead of union is A326881.

- The unlabeled case is A108798.

Cf. A003465, A072447, A102895, A102897, A108800, A193674, A193675, A306445, A326870, A326880, A326883.

Sequence in context: A082765 A132873 A244753 * A132552 A189273 A288554

Adjacent sequences:  A102891 A102892 A102893 * A102895 A102896 A102897

nonn,hard,more

Mitch Harris, Jan 18 2005

Additional comments from Don Knuth, Jul 01 2005

approved