Algebras with inner MB-representation.

Abstract

We investigate algebras of sets, and pairs $(\mathcal{A , I})$ consisting of an algebra $\mathcal{A}$ and an ideal $\mathcal{I} \subset \mathcal{A}$, that possess an inner MB-representation. We compare inner MB-representability of $(\mathcal{A , I})$ with several properties of $(\mathcal{A , I})$ considered by Baldwin. We show that $\mathcal{A}$ is inner MB-representable if and only if $\mathcal{A} =S(\mathcal{A} \setminus\mathcal{H}(\mathcal{A}))$, where $S(\cdot)$ is a Marczewski operation defined below and $\mathcal H$ consists of sets that are hereditarily in $\mathcal{A}$. We study the question of uniqueness of the ideal in that representation..