Algebra universalis

, Volume 71, Issue 1, pp 23–29 | Cite as

Large free sets in powers of universal algebras

  • Taras Banakh
  • Artur Bartoszewicz
  • Szymon Gła̧b
Open Access


We prove that for each universal algebra \({(A, \mathcal{A})}\) of cardinality \({|A| \geq 2}\) and infinite set X of cardinality \({|X| \geq | \mathcal{A}|}\) , the X-th power \({(A^{X}, \mathcal{A}^{X})}\) of the algebra \({(A, \mathcal{A})}\) contains a free subset \({\mathcal{F} \subset A^{X}}\) of cardinality \({|\mathcal{F}| = 2^{|X|}}\) . This generalizes the classical Fichtenholtz–Kantorovitch–Hausdorff result on the existence of an independent family \({\mathcal{I} \subset \mathcal{P}(X)}\) of cardinality \({|\mathcal{I}| = |\mathcal{P}(X)|}\) in the Boolean algebra \({\mathcal{P}(X)}\) of subsets of an infinite set X.

2010 Mathematics Subject Classification

Primary: 17A50 Secondary: 08A99 

Key words and phrases

free set universal algebra 


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Copyright information

© The Author(s) 2013

Open AccessThis article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Authors and Affiliations

  • Taras Banakh
    • 1
    • 2
  • Artur Bartoszewicz
    • 3
  • Szymon Gła̧b
    • 3
  1. 1.Ivan Franko University of LvivLvivUkraine
  2. 2.Jan Kochanowski UniversityKielcePoland
  3. 3.Institute of MathematicsTechnical University of ŁódźŁódźPoland

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