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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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

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
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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