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Bounds of finite relations

Part of the Lecture Notes in Mathematics book series (LNM,volume 560)

Abstract

Let \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{A}\)be a finite set of relations of degree n. A number N is found such that if \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{A}\)has cardinality less than N then for every relation R of degree n the following holds: if no member of \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{A}\)is embeddable in R then R has a strict extension in which no member of \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{A}\)is embeddable. In fact, if all members of \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{A}\)are defined on sets with at least n elements this number N is 3s(n,2). \(\mathop {II}\limits_{k = 3}^n\) 2s(n,k).k-1 where s(n,k) are the Stirling numbers of the second kind.

Keywords

  • Equivalence Class
  • Equivalence Relation
  • Binary Relation
  • Distinct Element
  • Identical Base

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. C. Berge, Principles of Combinatorics. (Academic Press, 1971)

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  2. R. Fraïssé, Course of Mathematical Logic, Vol. 1. (D. Reidel Publishing Company, 1973)

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© 1976 Springer-Verlag

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Gillam, D.W.H. (1976). Bounds of finite relations. In: Casse, L.R.A., Wallis, W.D. (eds) Combinatorial Mathematics IV. Lecture Notes in Mathematics, vol 560. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0097368

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  • DOI: https://doi.org/10.1007/BFb0097368

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-08053-4

  • Online ISBN: 978-3-540-37537-1

  • eBook Packages: Springer Book Archive