## Abstract

Let script A sign_{1}, . . ., script A sign_{n} script A sign_{n+1} be a finite sequence of algebras of sets given on a set X, ∪^{n}_{k=1} script A sign_{k} ≠ B-frakfur sign (X), with more than 4/3n pairwise disjoint sets not belonging to script A sign_{n+1}. It was shown in [4] and [5] that in this case ∪_{k-1}^{n+1} script A sign_{k} ≠ B-frakfur sign(X). Let us consider, instead script A sign_{n+1}, a finite sequence of algebras script A sign_{n+1}, . . . , script A sign_{n+l}. It turns out that if for each natural i ≤ l there exist no less than 4/3(n + l) - l/24 pairwise disjoint sets not belonging to script A sign_{n+i}, then ∪_{k=1}^{n+1} script A sign_{k} ≠ B-frakfur sign(X). But if l ≥ 195 and if for each natural i ≤ l there exist no less than 4/3(n + l) - l/15 pairwise disjoint sets not belonging to script A sign_{n+i}, then ∪_{k=1}^{n+l}script A sign _{k} ≠ B-frakfur sign(X). After consideration of finite sequences of algebras, it is natural to consider countable sequences of algebras. We obtained two essentially important theorems on a countable sequence of almost σ-algebras (the concept of almost σ -algebra was introduced in [4]).

Original language | English |
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Pages (from-to) | 483-500 |

Number of pages | 18 |

Journal | Journal of Symbolic Logic |

Volume | 72 |

Issue number | 2 |

DOIs | |

State | Published - Jun 2007 |

## Keywords

- Algebra of sets
- Almost σ -algebra
- Pairwise disjoint sets
- Ultrafilter