Dependence spaces

TL;DR

This paper provides a new proof of Steinitz's exchange theorem for infinite bases using the Kuratowski-Zorn Maximum Principle, expanding its applicability in dependence spaces and algebraic contexts.

Contribution

It introduces an alternative proof of Steinitz's exchange theorem for infinite bases without well-ordering, and explores dependence spaces with potential applications in algebra.

Findings

New proof of Steinitz's exchange theorem for infinite bases

Examples of dependence spaces with applications in algebra

Use of Kuratowski-Zorn Maximum Principle in proof

Abstract

The Steinitz exchange lemma is a basic theorem in linear algebra used, for example, to show that any two bases for a finite-dimensional vector space have the same number of elements. The result is named after the German mathematician Ernst Steinitz. We present another proof of the result of N.J.S. Hughes \cite{1} on Steinitz' exchange theorem for infinite bases. In our proof we assume Kuratowski-Zorn Maximum Principle instead of well ordering. We present some examples of dependence spaces of general nature with theirs possible applications of the result in other as linear or universal algebra domains of mathematical sciences. The lecture was presented on 77th Workshop on General Algebra, 24th Conference for Young Algebraists in Potsdam (Germany) on 21st March 2009.