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Large free sets in powers of universal algebras
Banakh, Taras, 1968- ; Bartoszewicz, Artur ; Głąb, Szymon
We prove that for each universal algebra ▫$(A, \mathcal{A})$▫ of cardinality ▫$\vert A \vert\geq 2$▫ and infinite set ▫$X$▫ of cardinality ▫$\vert X \vert \geq \vert \mathcal{A} \vert$▫, 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 ▫$\ver t\mathcal{F} \vert = 2^{\vert X \vert}$▫ . This generalizes the classical Fichtenholtz-Kantorovitch-Hausdorff result on the existence of an independent family ▫$\mathcal{I} \subset \mathcal{P}(X)$▫ of cardinality ▫$\vert\mathcal{I}\vert =\vert\mathcal{P}(X)\vert$▫ in the Boolean algebra ▫$\mathcal{P}(X)$▫ of subsets of an infinite set ▫$X$▫.
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