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

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- by Kyriakos Keremedis
- Proc. Amer. Math. Soc.
**124**(1996), 2527-2531 - DOI: https://doi.org/10.1090/S0002-9939-96-03305-9
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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*$\mathcal {A}=\{A_i:i\in k\}$

*of non empty sets there exists a family*$\mathcal {F=}\{F_i:i\in k\}$

*of odd sized sets such that, for every*$i\in k$, $F_i\subseteq A$ iff

*in every vector space*$B$

*over the two-element field every subspace*$V\subseteq B$

*has a complementary subspace*$S$ iff

*every quotient group of an abelian group each of whose elements has order 2 has a set of representatives.*

## References

- 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**, DOI 10.1090/conm/031/763890 - M. N. Bleicher,
*Some theorems on vector spaces and the axiom of choice*, Fund. Math.**54**(1964), 95–107. MR**164899**, DOI 10.4064/fm-54-1-95-107 - James D. Halpern,
*Bases in vector spaces and the axiom of choice*, Proc. Amer. Math. Soc.**17**(1966), 670–673. MR**194340**, DOI 10.1090/S0002-9939-1966-0194340-1 - Thomas J. Jech,
*The axiom of choice*, Studies in Logic and the Foundations of Mathematics, Vol. 75, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. MR**0396271** - 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**

## Bibliographic Information

**Kyriakos Keremedis**- Affiliation: University of the Aegean, Department of Mathematics, Karlovasi 83200, Samos, Greece
- Email: kker@kerkis.aegean.gr
- Received by editor(s): June 21, 1993
- Received by editor(s) in revised form: February 16, 1995
- Communicated by: Andreas R. Blass
- © Copyright 1996 American Mathematical Society
- 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