cp's OEIS Frontend

A102896 Number of ACI algebras (or semilattices) on n generators with no annihilator.

%I A102896 #62 Jul 13 2025 22:44:56
%S A102896 1,2,7,61,2480,1385552,75973751474,14087648235707352472
%N A102896 Number of ACI algebras (or semilattices) on n generators with no annihilator.
%C A102896 Or, number of Moore families on an n-set, that is, families of subsets that contain the universal set {1,...,n} and are closed under intersection.
%C A102896 Or, number of closure operators on a set of n elements.
%C A102896 An ACI algebra or semilattice is a system with a single binary, idempotent, commutative and associative operation.
%C A102896 Also the number of set-systems on n vertices that are closed under union. The BII-numbers of these set-systems are given by A326875. - _Gus Wiseman_, Jul 31 2019
%C A102896 From _Bernhard Ganter_, Jul 08 2025: (Start)
%C A102896 Also the number of union-free families of subsets of an n-set; i.e., families of nonempty sets on n elements such that no set is a union of some others.
%C A102896 Also the number of intersection-free families of subsets of an n-set; i.e., of families of proper subsets on n elements such that no set is an intersection of some others.
%C A102896 (Note that every union-free family on an n-set is the set of union-irreducible elements of exactly one union-closed family, and each family of union-irreducible elements is union-free. Same for intersection.) (End)
%D A102896 G. Birkhoff, Lattice Theory. American Mathematical Society, Colloquium Publications, Vol. 25, 3rd ed., Providence, RI, 1967.
%D A102896 Maria Paola Bonacina and Nachum Dershowitz, Canonical Inference for Implicational Systems, in Automated Reasoning, Lecture Notes in Computer Science, Volume 5195/2008, Springer-Verlag.
%D A102896 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 A102896 E. H. Moore, Introduction to a Form of General Analysis, AMS Colloquium Publication 2 (1910), pp. 53-80.
%H A102896 Andrew J. Blumberg, Michael A. Hill, Kyle Ormsby, Angélica M. Osorno, and Constanze Roitzheim, <a href="https://doi.org/10.1090/noti2882">Homotopical Combinatorics</a>, Notices Amer. Math. Soc. (2024) Vol. 71, No. 2, 260-266. See p. 261.
%H A102896 Daniel Borchmann and Bernhard Ganter, <a href="https://doi.org/10.1007/978-3-642-01815-2_2">Concept Lattice Orbifolds - First Steps</a>, Proceedings of the 7th International Conference on Formal Concept Analysis (ICFCA 2009), 22-37.
%H A102896 Jishnu Bose, Tien Chih, Hannah Housden, Legrand Jones II, Chloe Lewis, Kyle Ormsby, and Millie Rose, <a href="https://arxiv.org/abs/2503.22883">Combinatorics of factorization systems on lattices</a>, arXiv:2503.22883 [math.CO], 2025. See p. 11.
%H A102896 Pierre Colomb, Alexis Irlande, Olivier Raynaud and Yoan Renaud, <a href="https://www.researchgate.net/publication/233841963_About_The_Recursive_Decomposition_of_the_lattice_of_co-Moore_Families">About the Recursive Decomposition of the lattice of co-Moore Families</a>, ICFCA 2011.
%H A102896 Pierre Colomb, Alexis Irlande, Olivier Raynaud, and Yoan Renaud, <a href="https://doi.org/10.1007/s10472-013-9362-x">Recursive decomposition tree of a Moore co-family and closure algorithm</a>, Annals of Mathematics and Artificial Intelligence, 2013, DOI 10.1007/s10472-013-9362-x.
%H A102896 Nachum Dershowitz, Mitchell A. Harris, and Guan-Shieng Huang, <a href="http://arxiv.org/abs/cs/0610054">Enumeration Problems Related to Ground Horn Theories</a>, arXiv:cs/0610054v2 [cs.LO], 2006-2008.
%H A102896 Michel Habib and Lhouari 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), 291-296.
%F A102896 a(n) = Sum_{k=0..n} C(n, k)*A102894(k), where C(n, k) is the binomial coefficient.
%F A102896 For asymptotics see A102897.
%F A102896 a(n) = A102897(n)/2. - _Gus Wiseman_, Jul 31 2019
%e A102896 From _Gus Wiseman_, Jul 31 2019: (Start)
%e A102896 The a(0) = 1 through a(2) = 7 set-systems closed under union:
%e A102896   {}  {}     {}
%e A102896       {{1}}  {{1}}
%e A102896              {{2}}
%e A102896              {{1,2}}
%e A102896              {{1},{1,2}}
%e A102896              {{2},{1,2}}
%e A102896              {{1},{2},{1,2}}
%e A102896 (End)
%t A102896 Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],SubsetQ[#,Union@@@Tuples[#,2]]&]],{n,0,3}] (* _Gus Wiseman_, Jul 31 2019 *)
%Y A102896 For set-systems closed under union:
%Y A102896 - The covering case is A102894.
%Y A102896 - The unlabeled case is A193674.
%Y A102896 - The case also closed under intersection is A306445.
%Y A102896 - Set-systems closed under overlapping union are A326866.
%Y A102896 - The BII-numbers of these set-systems are given by A326875.
%Y A102896 Cf. A102895, A102897, A108798, A108800, A193675, A000798, A014466, A326878, A326880, A326881.
%K A102896 nonn,hard,more
%O A102896 0,2
%A A102896 _Mitch Harris_, Jan 18 2005
%E A102896 _N. J. A. Sloane_ added a(6) from the Habib et al. reference, May 26 2005
%E A102896 Additional comments from _Don Knuth_, Jul 01 2005
%E A102896 a(7) from Pierre Colomb (pierre(AT)colomb.me), Sep 04 2010