A193674 -id:A193674 - cp's OEIS frontend

A108799 Erroneous version of A193674.

Original entry on oeis.org

1, 2, 5, 19, 184, 14664, 108295801

Offset: 0

N. J. A. Sloane, Aug 02 2011

dead

The OEIS policy is to include incorrect published sequences, giving a pointer to the correct version.

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

Original entry on oeis.org

1, 2, 7, 61, 2480, 1385552, 75973751474, 14087648235707352472

Offset: 0

Mitch Harris, Jan 18 2005

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.
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 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
From Bernhard Ganter, Jul 08 2025: (Start)
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.
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.
(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)

			From _Gus Wiseman_, Jul 31 2019: (Start)
The a(0) = 1 through a(2) = 7 set-systems closed under union:
  {}  {}     {}
      {{1}}  {{1}}
             {{2}}
             {{1,2}}
             {{1},{1,2}}
             {{2},{1,2}}
             {{1},{2},{1,2}}
(End)

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). [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. 53-80.

Andrew J. Blumberg, Michael A. Hill, Kyle Ormsby, Angélica M. Osorno, and Constanze Roitzheim, Homotopical Combinatorics, Notices Amer. Math. Soc. (2024) Vol. 71, No. 2, 260-266. See p. 261.
Daniel Borchmann and Bernhard Ganter, Concept Lattice Orbifolds - First Steps, Proceedings of the 7th International Conference on Formal Concept Analysis (ICFCA 2009), 22-37.
Jishnu Bose, Tien Chih, Hannah Housden, Legrand Jones II, Chloe Lewis, Kyle Ormsby, and Millie Rose, Combinatorics of factorization systems on lattices, arXiv:2503.22883 [math.CO], 2025. See p. 11.
Pierre Colomb, Alexis Irlande, Olivier Raynaud and Yoan Renaud, About the Recursive Decomposition of the lattice of co-Moore Families, ICFCA 2011.
Pierre Colomb, Alexis Irlande, Olivier Raynaud, and Yoan Renaud, Recursive decomposition tree of a Moore co-family and closure algorithm, Annals of Mathematics and Artificial Intelligence, 2013, DOI 10.1007/s10472-013-9362-x.
Nachum Dershowitz, Mitchell A. Harris, and Guan-Shieng Huang, Enumeration Problems Related to Ground Horn Theories, arXiv:cs/0610054v2 [cs.LO], 2006-2008.
Michel Habib and Lhouari Nourine, The number of Moore families on n = 6, Discrete Math., 294 (2005), 291-296.
Caoilainn Kirkpatrick, Amelie el Mahmoud, Kyle Ormsby, Angélica M. Osorno, Dale Schandelmeier-Lynch, Riley Shahar, Lixing Yi, Avery Young, and Saron Zhu, Enumerating submonoids of finite commutative monoids, arXiv:2508.20786 [math.CO], 2025. See p. 16.

For set-systems closed under union:
- The covering case is A102894.
- The unlabeled case is A193674.
- The case also closed under intersection is A306445.
- Set-systems closed under overlapping union are A326866.
- The BII-numbers of these set-systems are given by A326875.
Cf. A102895, A102897, A108798, A108800, A193675, A000798, A014466, A326878, A326880, A326881.

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

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

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

A001930 Number of topologies, or transitive digraphs with n unlabeled nodes.

Original entry on oeis.org

1, 1, 3, 9, 33, 139, 718, 4535, 35979, 363083, 4717687, 79501654, 1744252509, 49872339897, 1856792610995, 89847422244493, 5637294117525695

Offset: 0

N. J. A. Sloane

			From _Gus Wiseman_, Aug 02 2019: (Start)
Non-isomorphic representatives of the a(0) = 1 through a(3) = 9 topologies:
  {}  {}{1}  {}{12}        {}{123}
             {}{2}{12}     {}{3}{123}
             {}{1}{2}{12}  {}{23}{123}
                           {}{1}{23}{123}
                           {}{3}{23}{123}
                           {}{2}{3}{23}{123}
                           {}{3}{13}{23}{123}
                           {}{2}{3}{13}{23}{123}
                           {}{1}{2}{3}{12}{13}{23}{123}
(End)

Loic Foissy, Claudia Malvenuto, Frederic Patras, Infinitesimal and B_infinity-algebras, finite spaces, and quasi-symmetric functions, Journal of Pure and Applied Algebra, Elsevier, 2016, 220 (6), pp. 2434-2458. .
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 218 (but the last entry is wrong).
M. Kolli, On the cardinality of the T_0-topologies on a finite set, Preprint, 2014.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
J. A. Wright, There are 718 6-point topologies, quasi-orderings and transgraphs, Notices Amer. Math. Soc., 17 (1970), p. 646, Abstract #70T-A106.
J. A. Wright, personal communication.
For further references concerning the enumeration of topologies and posets see under A000112 and A001035.

C. M. Bender et al., Combinatorics and field theory, arXiv:quant-ph/0604164, 2006.
Moussa Benoumhani, The Number of Topologies on a Finite Set, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.6.
M. Benoumhani and M. Kolli, Finite topologies and partitions, JIS 13 (2010) # 10.3.5
Gunnar Brinkmann and Brendan D. McKay, Counting unlabeled topologies and transitive relations.
G. Brinkmann and B. D. McKay, Counting unlabeled topologies and transitive relations, J. Integer Sequences, Volume 8, 2005.
Gunnar Brinkmann and Brendan D. McKay, Counting Unlabelled Topologies and Transitive Relations, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.1.
K. K.-H. Butler and G. Markowsky, Enumeration of finite topologies, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 169-184
K. K.-H. Butler and G. Markowsky, Enumeration of finite topologies, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 169-184. [Annotated scan of pages 180 and 183 only]
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
S. R. Finch, Transitive relations, topologies and partial orders
S. R. Finch, Transitive relations, topologies and partial orders, June 5, 2003. [Cached copy, with permission of the author]
L. Foissy, C. Malvenuto, and F. Patras, B_infinity-algebras, their enveloping algebras, and finite spaces, arXiv preprint arXiv:1403.7488 [math.AT], 2014-2015.
Misha Gavrilovich and Misha Rabinovich, The Quillen negation monoid of a category, and Schreier graphs of its action on classes of morphisms, 2024. See p. 11.
Dongseok Kim, Young Soo Kwon and Jaeun Lee, Enumerations of finite topologies associated with a finite graph, arXiv preprint arXiv:1206.0550, 2012. - From _N. J. A. Sloane_, Nov 09 2012
Messaoud Kolli, Direct and Elementary Approach to Enumerate Topologies on a Finite Set, J. Integer Sequences, Volume 10, 2007, Article 07.3.1.
G. Pfeiffer, Counting Transitive Relations, preprint, 2004.
G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
D. Rusin, Further information and references [Broken link]
D. Rusin, Further information and references [Cached copy]
Henry Sharp, Jr., Quasi-orderings and topologies on finite sets, Proceedings of the American Mathematical Society 17.6 (1966): 1344-1349. [Annotated scanned copy]
N. J. A. Sloane, List of sequences related to partial orders, circa 1972
N. J. A. Sloane, Classic Sequences
Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 8 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Eric Swartz and Nicholas J. Werner, Zero pattern matrix rings, reachable pairs in digraphs, and Sharp's topological invariant tau, arXiv:1709.05390 [math.CO], 2017.
J. M. Tangen and N. J. A. Sloane, Correspondence, 1976-1976
R. H. Warren, The number of topologies, Houston J. Math., 8 (No. 2, 1982), 297-301. Mentions a(4)=33. [Annotated scanned copy]
Eric Weisstein's World of Mathematics, Digraph Topology.
R. H. Warren, The number of topologies, Houston J. Math., 8 (No. 2, 1982), 297-301. Mentions a(4)=33. [Annotated scanned copy]
Wikipedia Topological space
J. A. Wright, There are 718 6-point topologies, quasiorderings and transgraphs, Preprint, 1970 [Annotated scanned copy]
J. A. Wright, Letter to N. J. A. Sloane, Apr 06 1972, listing 18 sequences

Cf. A000798 (labeled topologies), A001035 (labeled posets), A001930 (unlabeled topologies), A000112 (unlabeled posets), A006057, A001928, A001929.
The case with unions only is A108798.
The case with intersections only is (also) A108798.
Partial sums are A326898 (the non-covering case).
Cf. A000612, A003180, A108800, A193674, A306445, A326876, A326878, A326882.

a(8)-a(12) from Goetz Pfeiffer (goetz.pfeiffer(AT)nuigalway.ie), Jan 21 2004

a(13)-a(16) from Brinkmann's and McKay's paper, sent by Vladeta Jovovic, Jan 04 2006

A102894 Number of ACI algebras or semilattices on n generators, with no identity or annihilator.

Original entry on oeis.org

1, 1, 4, 45, 2271, 1373701, 75965474236, 14087647703920103947

Offset: 0

Mitch Harris, Jan 18 2005

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
Number of strict closure operators on a set of n elements, where the closure operator is said to be strict if the empty set is closed. - Tian Vlasic, Jul 30 2022

			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)

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.
E. H. Moore, Introduction to a Form of General Analysis, AMS Colloquium Publication 2 (1910), pp. 53-80.

Maria Paola Bonacina and Nachum Dershowitz, Canonical ground Horn theories, Lecture Notes in Computer Science 7797, 35-71 (2013).
P. Colomb, A. Irlande and O. Raynaud, Counting of Moore Families for n=7, International Conference on Formal Concept Analysis (2010).
N. Dershowitz, G. S. Huang and M. Harris, Enumeration Problems Related to Ground Horn Theories, arXiv:cs/0610054v2 [cs.LO], 2006-2008.
Christopher S. Flippen, Minimal Sets, Union-Closed Families, and Frankl's Conjecture, Master's thesis, Virginia Commonwealth Univ., 2023.
M. Habib and L. Nourine, The number of Moore families on n = 6, Discrete Math., 294 (2005), 291-296.

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.

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

Inverse binomial transform of A102896.

For asymptotics see A102897.

Additional comments from Don Knuth, Jul 01 2005

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

Original entry on oeis.org

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

Offset: 0

Mitch Harris, Jan 18 2005

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.

			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)

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.

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.

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.

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

For asymptotics see A102897.

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

Additional comments from Don Knuth, Jul 01 2005

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

A102897 Number of ACI algebras (or semilattices) on n generators.

Original entry on oeis.org

2, 4, 14, 122, 4960, 2771104, 151947502948, 28175296471414704944

Offset: 0

Mitch Harris, Jan 18 2005

Also counts Horn functions on n variables, Boolean functions whose set of truth assignments are closed under 'and', or equivalently, the Boolean functions that can be written as a conjunction of Horn clauses, clauses with at most one negative literal.
Also, number of families of subsets of {1,...,n} that are closed under intersection (because we can throw in the universe, or take it out, without affecting anything else).
An ACI algebra or semilattice is a system with a single binary, idempotent, commutative and associative operation.
Also the number of finite sets of finite subsets of {1..n} that are closed under union. - Gus Wiseman, Aug 03 2019

			a(2) = 14: Let the points be labeled a, b. We want the number of collections of subsets of {a, b} which are closed under intersection. 0 subsets: 1 way ({}), 1 subset: 4 ways (0; a; b; ab), 2 subsets: 5 ways (0,a; 0,b; 0,ab; a,ab; b,ab) [not a,b because their intersection, 0, would be missing], 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 14.
From _Gus Wiseman_, Aug 03 2019: (Start)
The a(0) = 2 through a(2) = 14 sets of subsets closed under union:
  {}    {}        {}
  {{}}  {{}}      {{}}
        {{1}}     {{1}}
        {{},{1}}  {{2}}
                  {{1,2}}
                  {{},{1}}
                  {{},{2}}
                  {{},{1,2}}
                  {{1},{1,2}}
                  {{2},{1,2}}
                  {{},{1},{1,2}}
                  {{},{2},{1,2}}
                  {{1},{2},{1,2}}
                  {{},{1},{2},{1,2}}
(End)

V. B. Alekseev, On the number of intersection semilattices [in Russian], Diskretnaya Mat. 1 (1989), 129-136.
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.
G. Burosch, J. Demetrovics, G. O. H. Katona, D. J. Kleitman and A. A. Sapozhenko, On the number of closure operations, in Combinatorics, Paul Erdős is Eighty (Volume 1), Keszthely: Bolyai Society Mathematical Studies, 1993, 91-105.
P. Colomb, A. Irlande and O. Raynaud, Counting of Moore Families for n=7, International Conference on Formal Concept Analysis (2010)
Alfred Horn, Journal of Symbolic Logic 16 (1951), 14-21. [See Lemma 7.]
D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.1, p. 79.
E. H. Moore, Introduction to a Form of General Analysis, AMS Colloquium Publication 2 (1910), pp. 53-80.

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

For nonempty set systems of the same type, see A121921.
Regarding sets of subsets closed under union:
- The case with an edge containing all of the vertices is A102895.
- The case without empty edges is A102896.
- The case with intersection instead of union is (also) A102897.
- The unlabeled version is A193675.
- The case closed under both union and intersection is A306445.
- The BII-numbers of set-systems closed under union are A326875.
- The covering case is A326906.
Cf. A102894, A108798, A108800, A193674, A193675, A326866, A326880, A326900.

Table[Length[Select[Subsets[Subsets[Range[n]]],SubsetQ[#,Union@@@Tuples[#,2]]&]],{n,0,3}] (* Gus Wiseman, Aug 03 2019 *)

a(n) = 2*A102896(n) = Sum_{k=0..n} C(n, k)*A102895(k), where C(n, k) is the binomial coefficient

Asymptotically, log_2 a(n) ~ binomial(n, floor(n/2)) for all of A102894, A102895, A102896 and this sequence [Alekseev; Burosch et al.]

Additional comments from Don Knuth, Jul 01 2005

A193675 Number of nonisomorphic systems enumerated by A102897; that is, the number of inequivalent Horn functions, under permutation of variables.

Original entry on oeis.org

2, 4, 10, 38, 368, 29328, 216591692, 5592326399531792

Offset: 0

Don Knuth, Jul 01 2005

When speaking of inequivalent Boolean functions, three groups of symmetries are typically considered: Complementations only, the Abelian group (2,...,2) of 2^n elements; permutations only, the symmetric group of n! elements; or both complementations and permutations, the octahedral group of 2^n n! elements. In this case only symmetry with respect to the symmetric group is appropriate because complementation affects the property of being a Horn function.

Also the number of non-isomorphic sets of subsets of {1..n} that are closed under union. - Gus Wiseman, Aug 04 2019

			From _Gus Wiseman_, Aug 04 2019: (Start)
Non-isomorphic representatives of the a(0) = 2 through a(2) = 10 sets of sets:
  {}    {}        {}
  {{}}  {{}}      {{}}
        {{1}}     {{1}}
        {{},{1}}  {{1,2}}
                  {{},{1}}
                  {{},{1,2}}
                  {{2},{1,2}}
                  {{},{2},{1,2}}
                  {{1},{2},{1,2}}
                  {{},{1},{2},{1,2}}
(End)

D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.1, p. 79.

P. Colomb, A. Irlande and O. Raynaud, Counting of Moore Families for n=7, International Conference on Formal Concept Analysis (2010).
D. E. Knuth, HORN-COUNT

The covering case is A326907.
The case without {} is A193674.
The labeled version is A102897.
The same with intersection instead of union is also A193675.
The case closed under both union and intersection also is A326908.
Cf. A102894, A102895, A102896, A102897, A108798, A108800, A326867, A326875, A326904.

a(n) = 2 * A193674(n).

a(6) received from Don Knuth, Aug 17 2005
a(6) corrected by Pierre Colomb, Aug 02 2011
a(7) = 2*A193674(7) from Hugo Pfoertner, Jun 18 2018

A108798 Number of nonisomorphic systems enumerated by A102894; that is, the number of inequivalent closure operators in which the empty set is closed. Also, the number of union-closed sets with n elements that contain the universe and the empty set.

Original entry on oeis.org

1, 1, 3, 14, 165, 14480, 108281182, 2796163091470050

Offset: 0

Don Knuth, Jul 01 2005

Also the number of unlabeled finite sets of subsets of {1..n} that contain {} and {1..n} and are closed under intersection. - Gus Wiseman, Aug 02 2019

			From _Gus Wiseman_, Aug 02 2019: (Start)
Non-isomorphic representatives of the a(0) = 1 through a(3) = 14 union-closed sets of sets:
  {}  {}{1}  {}{12}        {}{123}
             {}{2}{12}     {}{3}{123}
             {}{1}{2}{12}  {}{23}{123}
                           {}{1}{23}{123}
                           {}{3}{23}{123}
                           {}{13}{23}{123}
                           {}{2}{3}{23}{123}
                           {}{2}{13}{23}{123}
                           {}{3}{13}{23}{123}
                           {}{12}{13}{23}{123}
                           {}{2}{3}{13}{23}{123}
                           {}{3}{12}{13}{23}{123}
                           {}{2}{3}{12}{13}{23}{123}
                           {}{1}{2}{3}{12}{13}{23}{123}
(End)

Maria Paola Bonacina and Nachum Dershowitz, Canonical ground Horn theories, Lecture Notes in Computer Science 7797, 35-71 (2013).
G. Brinkmann and R. Deklerck, Generation of Union-Closed Sets and Moore Families, Journal of Integer Sequences, Vol.21 (2018), Article 18.1.7.
G. Brinkmann and R. Deklerck, Generation of Union-Closed Sets and Moore Families, arXiv:1701.03751 [math.CO], 2017.
Christopher S. Flippen, Minimal Sets, Union-Closed Families, and Frankl's Conjecture, Master's thesis, Virginia Commonwealth Univ., 2023.

The labeled version is A102894.

Cf. A000612, A001930, A003180, A102895, A102897, A108800, A193674, A193675, A326867, A326869, A326883.

a(n) = A108800(n)/2.

a(6) added (using A193674) by N. J. A. Sloane, Aug 02 2011

Added a(7), and reference to union-closed sets. - Gunnar Brinkmann, Feb 05 2018

A108800 Number of nonisomorphic systems enumerated by A102895.

Original entry on oeis.org

1, 2, 6, 28, 330, 28960, 216562364, 5592326182940100

Offset: 0

Don Knuth, Jul 01 2005

Also the number of non-isomorphic sets of sets with {} that are closed under intersection. Also the number of non-isomorphic set-systems (without {}) covering n + 1 vertices and closed under intersection. - Gus Wiseman, Aug 05 2019

			From _Gus Wiseman_, Aug 02 2019: (Start)
Non-isomorphic representatives of the a(0) = 1 through a(3) = 28 sets of sets with {} that are closed under intersection:
  {}  {}     {}            {}
      {}{1}  {}{1}         {}{1}
             {}{12}        {}{12}
             {}{1}{2}      {}{123}
             {}{2}{12}     {}{1}{2}
             {}{1}{2}{12}  {}{1}{23}
                           {}{2}{12}
                           {}{3}{123}
                           {}{1}{2}{3}
                           {}{23}{123}
                           {}{1}{2}{12}
                           {}{1}{3}{23}
                           {}{2}{3}{123}
                           {}{3}{13}{23}
                           {}{1}{23}{123}
                           {}{3}{23}{123}
                           {}{1}{2}{3}{23}
                           {}{1}{2}{3}{123}
                           {}{2}{3}{13}{23}
                           {}{1}{3}{23}{123}
                           {}{2}{3}{23}{123}
                           {}{3}{13}{23}{123}
                           {}{1}{2}{3}{13}{23}
                           {}{1}{2}{3}{23}{123}
                           {}{2}{3}{13}{23}{123}
                           {}{1}{2}{3}{12}{13}{23}
                           {}{1}{2}{3}{13}{23}{123}
                           {}{1}{2}{3}{12}{13}{23}{123}
(End)

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

Except a(0) = 1, first differences of A193675.
The connected case (i.e., with maximum) is A108798.
The same for union instead of intersection is (also) A108798.
The labeled version is A102895.
The case also closed under union is A326898.
The covering case is A326883.
Cf. A001930, A102894, A102896, A102897, A193674, A326880, A326881.

a(n > 0) = 2 * A108798(n).

a(6) added (using A193675) by N. J. A. Sloane, Aug 02 2011
Changed a(0) from 2 to 1 by Gus Wiseman, Aug 02 2019
a(7) added (using A108798) by Andrew Howroyd, Aug 10 2019

A326881 Number of set-systems with {} that are closed under intersection and cover n vertices.

Original entry on oeis.org

1, 1, 5, 71, 4223, 2725521, 151914530499, 28175294344381108057

Offset: 0

Gus Wiseman, Jul 30 2019

			The a(2) = 5 set-systems:
  {{},{1,2}}
  {{},{1},{2}}
  {{},{1},{1,2}}
  {{},{2},{1,2}}
  {{},{1},{2},{1,2}}

The case also closed under union is A000798.
The connected case (i.e., with maximum) is A102894.
The same for union instead of intersection is (also) A102894.
The non-covering case is A102895.
The BII-numbers of these set-systems (without the empty set) are A326880.
The unlabeled case is A326883.
Cf. A003465, A014466, A102896, A102897, A193674, A193675, A306445, A307249, A326878.

Table[Length[Select[Subsets[Subsets[Range[n]]],MemberQ[#,{}]&&Union@@#==Range[n]&&SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}]

Inverse binomial transform of A102895. - Andrew Howroyd, Aug 10 2019

a(5)-a(7) from Andrew Howroyd, Aug 10 2019

cp's OEIS Frontend

A108799 Erroneous version of A193674.

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

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A001930 Number of topologies, or transitive digraphs with n unlabeled nodes.

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A102894 Number of ACI algebras or semilattices on n generators, with no identity or annihilator.

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

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A102897 Number of ACI algebras (or semilattices) on n generators.

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A193675 Number of nonisomorphic systems enumerated by A102897; that is, the number of inequivalent Horn functions, under permutation of variables.

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A108798 Number of nonisomorphic systems enumerated by A102894; that is, the number of inequivalent closure operators in which the empty set is closed. Also, the number of union-closed sets with n elements that contain the universe and the empty set.

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