Abstract
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.
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Presented by A. Dow.
The first and third authors have been supported by the NCN grant DEC- 2012/07/D/ST1/02087. The second and the third authors have been supported by the Polish Ministry of Science and Higher Education Grant No N N201 414939 (2010-2013).
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Banakh, T., Bartoszewicz, A. & Gła̧b, S. Large free sets in powers of universal algebras. Algebra Univers. 71, 23–29 (2014). https://doi.org/10.1007/s00012-013-0261-0
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DOI: https://doi.org/10.1007/s00012-013-0261-0