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One-transversal families

Published online by Cambridge University Press:  24 October 2008

Richard Rado
Affiliation:
University of Reading

Extract

Let I be a set and

be a family of sets Aν labelled by the elements of I. Throughout, parentheses ( ) denote families and curly brackets {} denote sets. A transversal, or system of distinct representatives, of F is, by definition, a family (xν: ν ∈ I) of objects xν, for ν ∈ I, such that †

Thus the family ({1, 2,} {1, 2}) has exactly two transversals, namely (1,2) and (2, 1). Let TF denote the set of all transversals of F. Much work has been done on the question of characterizing those F for which TF ≠ Ø. We are here going to characterize those F for which TF has exactly one element. In contrast to the more familiar case TF ≠ Ø, our argument is effective irrespective of the cardinalities of I and Aν.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1975

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References

REFERENCES

(1)Steffens, K. P. Anzahluntersuchungen von Transversalen und injektiven Auswahlfunktionen in abzählbaren Familien mit lauter endlichen Mitgliedern. Schriften zur Informatik und Angewandten Mathematik, 12 1973, Bericht no. 10. Rheinisch-Westfälische Technische Hochschule, Aachen.Google Scholar
(2)Manaster, A. B. and Rosenstein, J. G.Effective matchmaking. Proc. London Math. Soc. 25 (1972), 615654.Google Scholar