Large free sets in universal algebras

TL;DR

This paper demonstrates that for any universal algebra of size at least two and an infinite set with sufficient cardinality, its power contains a free subset of maximal possible size, generalizing classical independence results.

Contribution

It generalizes the classical independence result to universal algebras, showing the existence of large free subsets in their powers.

Findings

Existence of free subsets of size 2^{|X|} in algebra powers

Generalization of classical independence in Boolean algebras

Applicable to universal algebras of size ≥ 2

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 $X$.