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On Sets Not Belonging to Algebras of Subsets
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Memoirs of the American Mathematical Society
1992; 111 pp; softcover
Volume: 100
ISBN-10: 0-8218-2541-0
ISBN-13: 978-0-8218-2541-9
List Price: US$31 Individual Members: US$18.60
Institutional Members: US\$24.80
Order Code: MEMO/100/480

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.

Combinatorists, set-theorists and general topologists.

Reviews

"A fascinating new angle that shows that non-measurable sets are here to stay."

-- The Bulletin of Mathematics Books and Computer Software

• Introduction
• Main results
• Fundamental idea
• Finite sequences of algebras (1)
• Countable sequences of algebras (1)
• Proof of Theorem II
• Improvement of Theorem II (Proof of Theorem II*)
• Proof of Theorems III and IV
• The inverse problem
• Finite sequences of algebras (2)
• Countable sequences of algebras (2)
• Improvement of some main results
• Sets not belonging to semi-lattices of subsets and not belonging to lattices of subsets
• Unsolved problems