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Minimum-sized Infinite Partitions of Boolean Algebras

✍ Scribed by J. Donald Monk

Publisher
John Wiley and Sons
Year
1996
Tongue
English
Weight
736 KB
Volume
42
Category
Article
ISSN
0044-3050

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✦ Synopsis

Abstract

For any Boolean Algebra A, let c~mm~(A) be the smallest size of an infinite partition of unity in A. The relationship of this function to the 21 common functions described in Monk [4] is described, for the class of all Boolean algebras, and also for its most important subclasses. This description involves three main results: the existence of a rigid tree algebra in which c~mm~ exceeds any preassigned number, a rigid interval algebra with that property, and the construction of an interval algebra in which every well‐ordered chain has size less than c~mm~.

Mathematics Subject Classification: 06E05, 03E05.

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Let 2 [n] denote the Boolean lattice of order n, that is, the poset of subsets of {1, ..., n} ordered by inclusion. Recall that 2 [n] may be partitioned into what we call the canonical symmetric chain decomposition (due to de Bruijn, Tengbergen, and Kruyswijk), or CSCD. Motivated by a question of FΓΌ