A solution of Dedekind's problem on the number of isotone Boolean functions.

@article{Kisielewicz1988ASO,
  title={A solution of Dedekind's problem on the number of isotone Boolean functions.},
  author={Andrzej P. Kisielewicz},
  journal={Journal f{\"u}r die reine und angewandte Mathematik (Crelles Journal)},
  year={1988},
  volume={1988},
  pages={139 - 144}
}
  • A. Kisielewicz
  • Published 1988
  • Mathematics
  • Journal für die reine und angewandte Mathematik (Crelles Journal)
Almost one hundred years ago R. Dedekind raised the problem to determine the number D (n) of elements in the free bounded distributive lattice on n generators which is the same s the number of isotone maps from the Boolean lattice T into 2. D (n) is also the number of antichains in 2 and also the number of simplicial complexes on n elements (cf. [4], [9], [20], [21]). The problem was considered by many authors, but so far only partial results were obtained: asymptotic estimations [12], [13… 
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References

SHOWING 1-10 OF 22 REFERENCES
On Dedekind’s problem: the number of isotone Boolean functions. II
It is shown that 0(n), the size of the free distributive lattice on n generators (which is the number of isotone Boolean functions on subsets of an n element set), satisfies [n1 i (n) < 2(1 +0(1og
On Dedekind’s problem: The number of monotone Boolean functions
with an = ce~nli, /3„ = e'(Iog w)/«1'2.The number \p(n) is equal to the number of ideals, or of antichains,or of monotone increasing functions into 0 and 1 definable on thelattice of subsets of an
Recursive formulas on free distributive lattices
Enumeration in Classes of Ordered Structures
Let K be a class of finite structures. There are two obvious enumeration questions to ask of K: what is the set of cardinalities of members of K (the spectrum problem) and, for each n ≥ 1 what is the
Über Zerlegungen von Zahlen Durch Ihre Grössten Gemeinsamen Theiler
Liegt ein endliches System von naturlichen Zahlen vor, und bildet man alle grossten gemeinsamen Theiler von zwei oder mehreren dieser Zahlen, so werden die letzteren hierdurch auf mannigfaltige Weise
Dedekind's Problem
  • Order 2
  • 1986
On the number of monotone Boolean functions
  • Probl. Kibern. 38
  • 1981
A kind of matrix representation of a free distributive lattice and evaluation of its order
  • Wuhan Daxue Xuebo
  • 1980
Solution of Dedekind's problem on the number of monotone Boolean functions
  • Soviet Math. Dokl. 18
  • 1977
hler, Cardinalities of finite distributive lattices
  • Mitt. Math. Sem. Gie en
  • 1976
...
1
2
3
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