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Independent Axioms for Vector Spaces

Published online by Cambridge University Press:  03 November 2016

J. F. Rigby  and
James Wiegold
J. F. Rigby
Affiliation:
Mathematical Institute, Hacettepe University, Ankara, Turkey
James Wiegold
Affiliation:
Dept. of Pure Mathematics, University College, Cathays Park, Cardiff
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Extract

In a recent paper [1], Victor Bryant shows how the number of axioms required to define a vector space can be reduced to seven (in addition to closure requirements). The main result of his article is that commutativity of addition can be deduced from the other axioms. In the present article we show how to reduce this number to six. For certain underlying fields one or more of these axioms can be deduced from the others. However, the six axioms are in general independent; we invite interested readers to show this by constructing their own counter-examples, which the editor of the Gazette will be pleased to receive.

Type
Research Article
Information
The Mathematical Gazette , Volume 57 , Issue 399 , February 1973 , pp. 56 - 62
Copyright
Copyright © Mathematical Association 1973

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References

1. Bryant, Victor, Reducing classical axioms, Mathl Gaz. LV, 38–40 (No. 391, February 1971).10.2307/3613304CrossRefGoogle Scholar
2. Kurosh, A. G., The Theory of Groups, Vol. I. Chelsea Publishing Co. (1955).Google Scholar
3. Liebeck, Hans, The vector space axiom lv = v, Mathl Gaz. LVI, 30–33 (No. 395, February 1972).10.2307/3613685CrossRefGoogle Scholar