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Minimal dependent sets

Published online by Cambridge University Press:  09 April 2009

D. S. Asche
D. S. Asche
Affiliation:
Department of Mathematics Monash University Clayton, Victoria
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The subject matter of this note is the notion of a dependence structure on an abstract set. There are a number of different approaches to this topic and it is known that many of these lead to precisely the same structure. Axioms are given here to specify the minimal dependent sets for such a structure. They are closely related to conditions introduced by Hassler Whitney in [1] and to a certain “elimination axiom” for the independent sets.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 6 , Issue 3 , August 1966 , pp. 259 - 262
Copyright
Copyright © Australian Mathematical Society 1966

References

[1]Whitney, Hassler, ‘On the abstract properties of linear dependence’, Amer. J. Math. 57 (1935), 509–533.CrossRefGoogle Scholar
[2]Robertson, A. P. and Weston, J. D., ‘A general basis theorem’, Proc. Edinburgh Math. Soc. (2) 11 (1958—1959), 139–141.CrossRefGoogle Scholar
[3]Rado, R., ‘Note on independence functions’, Proc. London Math. Soc. 7 (1957), 300–320.CrossRefGoogle Scholar
[4]Ingleton, A. W., ‘A note on independence functions and rank’, J. London Math. Soc. 34 (1959), 49–56.CrossRefGoogle Scholar