A note on well-generated Boolean algebras in models satisfying Martin's axiom

R. Bonnet, M. Rubin

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


A Boolean algebra B is well generated, if it has a well-founded sublattice L such that L generates B. Let B be a superatomic Boolean algebra. The rank of B (rk(B)) is defined to be the Cantor Bendixon rank of the Stone space X of B. For every i≤rk(B) let λi(B) be the number of isolated points in the i's Cantor Bendixon derivative of X. The cardinal sequence of B is defined as λ→(B):〈λi(B):i≤rk(X)〉. If a∈B-{0}, then the rank of a (rk(a)) is defined as the rank of the Boolean algebra B⌈a:{b∈B:b≤a}. An element a∈B-{0} is a generalized atom (a∈At(B)), if the last cardinal in the cardinal sequence of B⌈a is 1. Let a,b∈At(B). We denote a∼Bb, if rk(a)=rk(b)=rk(a·b). A subset H⊆At(B) is a complete set of representatives (CSR) for B, if for every a∈At(B) there is a unique b∈H such that b∼Ba. Any CSR for B generates B. We say that B is hereditarily decreasingly canonically well generated, if for every subalgebra C of B and every CSR H for C there is a CSR M for C such that: (1) for every a∈H and b∈M: if b∼Ca then b≤a; (2) the sublattice of C generated by M is well founded. Theorem. Assume (MA+א1<2א0). Let ε be a countable ordinal, κ<2א0 and n<ω. If B is a superatomic Boolean algebra such that λ→(B)=〈א0:i<ε〉〈κ, א1,n〉 or λ→(B)=〈א0,2א0,א1,n〉, then B is hereditarily decreasingly canonically well generated.

Original languageEnglish
Pages (from-to)7-18
Number of pages12
JournalDiscrete Mathematics
Issue number1-3
StatePublished - 6 Mar 2005


  • Consistency result
  • Superatomic Boolean algebras
  • Well-founded lattices

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics


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