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On essentially low, canonically well-generated Boolean algebras

Published online by Cambridge University Press:  12 March 2014

Robert Bonnet  and
Matatyahu Rubin
Robert Bonnet
Affiliation:
Laboratoire De Mathématiques, Université De Savoie, Le Bourget-Du-Lac, France, E-mail: bonnet@in2p3.fr
Matatyahu Rubin
Affiliation:
Department of Mathematics and CS, Ben Gurion University, Beer Sheva, Israel, E-mail: matti@math.bgu.ac.il
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Abstract

Let B be a superatomic Boolean algebra (BA). The rank of B (rk(B)). is defined to be the Cantor Bendixon rank of the Stone space of B. If a ∈ B − {0}, then the rank of a in B (rk(a)). is defined to be the rank of the Boolean algebra . The rank of 0B is defined to be −1. An element a ∈ B − {0} is a generalized atom , if the last nonzero cardinal in the cardinal sequence of B ↾ a is 1. Let a, b ∈ . We denote a ˜ b, if rk(a) = rk(b) = rk(a · b). A subset H ⊆ is a complete set of representatives (CSR) for B, if for every a there is a unique h ∈ H such that h ~ a. Any CSR for B generates B. We say that B is canonically well-generated (CWG), if it has a CSR H such that the sublattice of B generated by H is well-founded. We say that B is well-generated, if it has a well-founded sublattice L such that L generates B.

Theorem 1. Let B be a Boolean algebra with cardinal sequence . If B is CWG, then every subalgebra of B is CWG.

A superatomic Boolean algebra B is essentially low (ESL), if it has a countable ideal I such that rk(B/I) ≤ 1.

Theorem 1 follows from Theorem 2.9. which is the main result of this work. For an ESL BA B we define a set FB of partial functions from a certain countably infinite set to ω (Definition 2.8). Theorem 2.9 says that if B is an ESL Boolean algebra, then the following are equivalent.

(1) Every subalgebra of B is CWG: and

(2) FB is bounded.

Theorem 2. If an ESL Boolean algebra is not CWG, then it has a subalgebra which is not well-generated.

Keywords

superatomic Boolean algebrasbounding number of ℘(ω)
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 67 , Issue 1 , March 2002 , pp. 369 - 396
Copyright
Copyright © Association for Symbolic Logic 2002

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References

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