# On sets not belonging to algebras of subsets

### About this Title

**L. Š. Grinblat**

Publication: Memoirs of the American Mathematical Society

Publication Year
1992: Volume 100, Number 480

ISBNs: 978-0-8218-2541-9 (print); 978-1-4704-0057-6 (online)

DOI: http://dx.doi.org/10.1090/memo/0480

MathSciNet review: 1124108

MSC: Primary 04A20; Secondary 03E05, 28A05, 54D35

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The main results of this work can be formulated in such
an elementary way that it is likely to attract mathematicians from
a broad spectrum of specialties, though its main audience will likely
be combintorialists, set-theorists, and topologists. The central
question is this: Suppose one is given an at most countable family of
algebras of subsets of some fixed set such that, for each algebra, there
exists at least one set that is not a member of that algebra. Can one
then assert that there is a set that is not a member of *any*
of the algebras? Although such a set clearly exists in the case of one
or two algebras, it is very easy to construct an example of three
algebras for which no such set can be found. Grinblat's principal
concern is to determine conditions that, if imposed on the algebras,
will insure the existence of a set not belonging to any of them. If the
given family of algebras is finite, one arrives at a purely
combinatorial problem for a finite set of ultrafilters. If the family is
countably infinite, however, one needs not only combinatorics of
ultrafilters but also set theory and general topology.

Readership

Combinatorists, set-theorists and
general topologists.

### Table of Contents

**Chapters**

- 1. Introduction
- 2. Main results
- 3. Fundamental idea
- 4. Finite sequences of algebras (1)
- 5. Countable sequences of algebras (1)
- 6. Proof of Theorem II
- 7. Improvement of Theorem II (proof of Theorem II*)
- 8. Proof of Theorems III and IV
- 9. The inverse problem
- 10. Finite sequences of algebras (2)
- 11. Countable sequences of algebras (2)
- 12. Improvement of some main results
- 13. Sets not belonging to semi-lattices of subsets and not belonging to
lattices of subsets
- 14. Unsolved problems