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Diskretnaya Matematika, 1989, Volume 1, Issue 2, Pages 129–136 (Mi dm915)  

This article is cited in 2 scientific papers (total in 2 papers)

The number of families of subsets that are closed with respect to intersections

V. B. Alekseev
Abstract: Let $\alpha(n)$ be the number of families of subsets of an $n$-element set having the property: for every two subsets of the family, their intersection belongs to the family as well. In this article it is proved that $\log_2\alpha(n)\sim C_n^{[n/2]}$ as $n\to\infty$. It follows from this that the same result is also valid for some other structures, in particular for the number of all possible closure operations on an $n$-element set. These results are obtained as a corollary of the analogous result as $n\to \infty$ and $k=o(\sqrt n/\log_2^2n)$ for the number of families of subsets of an $n$-element set which satisfy the condition: if $k$ one-element extensions of a subset $A$ belong to the family, then $A$ belongs to the family as well. Since there is a correspondence between families of subsets and functions of logic algebra, these results establish also asymptotics of the logarithm for the number of functions of the logic algebra of $n$ variables with the corresponding properties.
Received: 15.11.1988
Bibliographic databases:
UDC: 510.716
Language: Russian
Citation: V. B. Alekseev, “The number of families of subsets that are closed with respect to intersections”, Diskr. Mat., 1:2 (1989), 129–136
Citation in format AMSBIB
\Bibitem{Ale89}
\by V.~B.~Alekseev
\paper The number of families of subsets that are closed with respect to intersections
\jour Diskr. Mat.
\yr 1989
\vol 1
\issue 2
\pages 129--136
\mathnet{http://mi.mathnet.ru/dm915}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1035100}
\zmath{https://zbmath.org/?q=an:0725.05009}
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  • https://www.mathnet.ru/eng/dm/v1/i2/p129
  • This publication is cited in the following 2 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Дискретная математика
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