Finite and countable families of algebras of sets

Let $\{\A_\la\}_{\la \in \La}$ be a family of algebras of sets defined on a set $X, 0 <\# (\La) \leq \aleph_0$, and $\A_\la \neq \P(X)$ for each $\la \in \La$. We assume that $\A_\la$ are $\sigma$-algebras if $\# (\La) = \aleph_0$. We obtained the necessary and sufficient conditions for which $\bigcup_{\la \in \La} \A_\la = \P(X)$. In the formulation of these conditions we use $\omega$-saturated algebras and finite sequences of ultrafilters on $X$.

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Published 1 January 2010