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Theorems on sets not belonging to algebras

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## Theorems on sets not belonging to algebras

### L. S. Grinblat

Abstract.
Let $\mathcal{A}_1,\dots, \mathcal{A}_n, \mathcal{A}_{n+1}$ be a finite sequence of algebras of sets given on a set $X$, $\bigcup_{k=1}^n \mathcal{A}_k \ne \mathfrak{P}(X)$, with more than $\frac{4}{3}n$ pairwise disjoint sets not belonging to $\mathcal{A}_{n+1}$. It has been shown in the author's previous articles that in this case $\bigcup_{k=1}^{n+1} \mathcal{A}_k \ne \mathfrak{P}(X)$. Let us consider, instead of $\mathcal{A}_{n+1}$, a finite sequence of algebras $\mathcal{A}_{n+1}, \dots, \mathcal{A}_{n+l}$. It turns out that if for each natural $i \le l$ there exist no less than $\frac{4}{3}(n+l)- \frac{l}{24}$ pairwise disjoint sets not belonging to $\mathcal{A}_{n+i}$, then $\bigcup_{k=1}^{n+l} \mathcal{A}_k \ne \mathfrak{P}(X)$. Besides this result, the article contains: an essentially important theorem on a countable sequence of almost $\sigma$-algebras (the concept of almost $\sigma$-algebra was introduced by the author in 1999), a theorem on a family of algebras of arbitrary cardinality (the proof of this theorem is based on the beautiful idea of Halmos and Vaughan from their proof of the theorem on systems of distinct representatives), a new upper estimate of the function $\mathfrak{v}(n)$ that was introduced by the author in 2002, and other new results.

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#### Article Info

• ERA Amer. Math. Soc. 10 (2004), pp. 51-57
• Publisher Identifier: S 1079-6762(04)00129-5
• 2000 Mathematics Subject Classification. Primary 03E05; Secondary 54D35
• Key words and phrases. Algebra on a set, almost $\sigma$-algebra, ultrafilter, pairwise disjoint sets
• Received by editors February 15, 2004
• Posted on May 26, 2004
• Communicated by David Kazhdan