Bases for vector spaces over the two-element field and the axiom of choice

Author:
Kyriakos Keremedis

Journal:
Proc. Amer. Math. Soc. **124** (1996), 2527-2531

MSC (1991):
Primary 03E25

DOI:
https://doi.org/10.1090/S0002-9939-96-03305-9

MathSciNet review:
1322930

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Abstract: It is shown that the axiom of choice follows in a weaker form than the Zermelo - Fraenkel set theory from the assertion: * every set of generators G of a vector space V over the two element field includes a basis L for V*. It is also shown that: * for every family * * of non empty sets there exists a family * * of odd sized sets such that, for every *, iff * in every vector space * * over the two-element field every subspace * * has a complementarysubspace * iff * every quotient group of an abelian group each of whose elements has order 2 has a set of representatives.*

**1.**A. Blass,*Existence of bases implies the axiom of choice*, in Axiomatic Set Theory, Contemporary Mathematics,**31**(1984) 31 - 33. MR**86a:04001****2.**M. Bleicher,*Some theorems on vector spaces and the axiom of choice*, Fund. Math.**54**(1964), 95 - 107. MR**29:2190****3.**J. D. Halpern,*Bases for vector spaces and the axiom of choice*, Proc. Amer. Math. Soc.,**17**(1966) 670 - 673. MR**33:2550****4.**T. Jech,*The axiom of choice*, North-Holland, 1973. MR**53:139****5.**H. Rubin and J.E. Rubin,*Equivalents of the Axiom of Choice*, II, North-Holland, 1985. MR**87c:04004**

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Additional Information

**Kyriakos Keremedis**

Affiliation:
University of the Aegean, Department of Mathematics, Karlovasi 83200, Samos, Greece

Email:
kker@kerkis.aegean.gr

DOI:
https://doi.org/10.1090/S0002-9939-96-03305-9

Received by editor(s):
June 21, 1993

Received by editor(s) in revised form:
February 16, 1995

Communicated by:
Andreas R. Blass

Article copyright:
© Copyright 1996
American Mathematical Society