## Abstract

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 language | English |
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Pages (from-to) | 7-18 |

Number of pages | 12 |

Journal | Discrete Mathematics |

Volume | 291 |

Issue number | 1-3 |

DOIs | |

State | Published - 6 Mar 2005 |

## Keywords

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

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics