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

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.**Andreas Blass,*Existence of bases implies the axiom of choice*, Axiomatic set theory (Boulder, Colo., 1983) Contemp. Math., vol. 31, Amer. Math. Soc., Providence, RI, 1984, pp. 31–33. MR**763890**, 10.1090/conm/031/763890**2.**M. N. Bleicher,*Some theorems on vector spaces and the axiom of choice*, Fund. Math.**54**(1964), 95–107. MR**0164899****3.**James D. Halpern,*Bases in vector spaces and the axiom of choice*, Proc. Amer. Math. Soc.**17**(1966), 670–673. MR**0194340**, 10.1090/S0002-9939-1966-0194340-1**4.**Thomas J. Jech,*The axiom of choice*, North-Holland Publishing Co., Amsterdam-London; Amercan Elsevier Publishing Co., Inc., New York, 1973. Studies in Logic and the Foundations of Mathematics, Vol. 75. MR**0396271****5.**Herman Rubin and Jean E. Rubin,*Equivalents of the axiom of choice. II*, Studies in Logic and the Foundations of Mathematics, vol. 116, North-Holland Publishing Co., Amsterdam, 1985. MR**798475**

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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