Counting Ω-ideals

Abstract

Let ${\mathbb{A}}$ be a universal algebra of signature Ω, and let ${\mathcal{I}}$ be an ideal in the Boolean algebra ${\mathcal{P}_{\mathbb{A}}}$ of all subsets of ${\mathbb{A}}$ . We say that ${\mathcal{I}}$ is an Ω-ideal if ${\mathcal{I}}$ contains all finite subsets of ${\mathbb{A}}$ and ${f(A^{n}) \in \mathcal{I}}$ for every n-ary operation ${f \in \Omega}$ and every ${A \in \mathcal{I}}$ . We prove that there are ${2^{2^{\aleph_0}}}$ Ω-ideals in ${\mathcal{P}_{\mathbb{A}}}$ provided that ${\mathbb{A}}$ is countably infinite and Ω is countable.