A102895 Number of ACI algebras or semilattices on n generators with no identity element.

1, 2, 8, 90, 4542, 2747402, 151930948472, 28175295407840207894

Offset: 0

Author

Mitch Harris, Jan 18 2005

Keywords

• nonn
• hard
• more

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.

Examples

			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.
From _Gus Wiseman_, Aug 02 2019: (Start)
The a(0) = 1 through a(2) = 8 sets of sets with {} that are closed under intersection are:
  {{}}  {{}}      {{}}
        {{},{1}}  {{},{1}}
                  {{},{2}}
                  {{},{1,2}}
                  {{},{1},{2}}
                  {{},{1},{1,2}}
                  {{},{2},{1,2}}
                  {{},{1},{2},{1,2}}
(End)
		

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, 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.

Links

• 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.

Crossrefs

The connected case (i.e., with maximum) is A102894.

The same for union instead of intersection is A102896.

The unlabeled version is A108800.

The case also closed under union is A326878.

The BII-numbers of these set-systems (without the empty set) are A326880.

The covering case is A326881.

Cf. A000798, A102897, A108798, A193674, A193675, A306445, A326883.

Programs

Mathematica

Table[Length[Select[Subsets[Subsets[Range[n]]],MemberQ[#,{}]&&SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}] (* Gus Wiseman, Aug 02 2019 *)

Formula

For asymptotics see A102897.

a(n > 0) = 2 * A102894(n).

Extensions

Additional comments from Don Knuth, Jul 01 2005

Changed a(0) from 2 to 1 by Gus Wiseman, Aug 02 2019