Large free sets in universal algebras

Abstract

We prove that for each universal algebra $(A,\mathcal{A})$ of cardinality $|A|\ge 2$ and an infinite set $X$ of cardinality $|X|\ge |\mathcal{A}|$, 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 $X$.

4 pages

References in corpus (1)

• Large free linear algebras of real and complex functions