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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 $\mathcal {A}=\{A_i:i\in k\}$ 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 complementarysubspace $S$ iff every quotient group of an abelian group each of whose elements has order 2 has a set of representatives.

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

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