TY - JOUR

T1 - Chains of well-generated Boolean algebras whose union is not well-generated

AU - Bonnet, Robert

AU - Rubin, Matatyahu

N1 - Funding Information:
* Supported by The Center For Advanced Studies in Mathematics "Ben Gurion University". Received September 30, 2004

PY - 2006/9/18

Y1 - 2006/9/18

N2 - A Boolean algebra B that has a well-founded sublattice which generates B is called a well-generated Boolean algebra. Every well-generated Boolean algebra is superatomic. However, there are superatomic algebras which are not well-generated. We consider two types of increasing chains of Boolean algebras, canonical chains and rank preserving chains, and show that the class of well-generated Boolean algebras is not closed under union of such chains, even when these chains are taken to be countable. A Boolean algebra is superatomic iff its Stone space is scattered. If B is superatomic and a ∈ B, then the rank of a is the Cantor Bendixon rank of the Stone space of {b} b ≤ a}. A chain {Bα α << δ} is a canonical chain if for every α < β <, δ, Bα is the subagebra of Bβ generated by all members of Bβ whose rank is < α. For a superatomic algebra B, I(B) denotes the ideal consisting of all members of B whose rank is less than the rank of B. A chain {Bα| α < δ} is a rank preserving chain if for every α < β < δ and α ∈ I(B α), the rank and mutiplicity of a in Bα are equal to the rank and mutiplicity of a in Bβ.

AB - A Boolean algebra B that has a well-founded sublattice which generates B is called a well-generated Boolean algebra. Every well-generated Boolean algebra is superatomic. However, there are superatomic algebras which are not well-generated. We consider two types of increasing chains of Boolean algebras, canonical chains and rank preserving chains, and show that the class of well-generated Boolean algebras is not closed under union of such chains, even when these chains are taken to be countable. A Boolean algebra is superatomic iff its Stone space is scattered. If B is superatomic and a ∈ B, then the rank of a is the Cantor Bendixon rank of the Stone space of {b} b ≤ a}. A chain {Bα α << δ} is a canonical chain if for every α < β <, δ, Bα is the subagebra of Bβ generated by all members of Bβ whose rank is < α. For a superatomic algebra B, I(B) denotes the ideal consisting of all members of B whose rank is less than the rank of B. A chain {Bα| α < δ} is a rank preserving chain if for every α < β < δ and α ∈ I(B α), the rank and mutiplicity of a in Bα are equal to the rank and mutiplicity of a in Bβ.

UR - http://www.scopus.com/inward/record.url?scp=33748570777&partnerID=8YFLogxK

U2 - 10.1007/BF02773602

DO - 10.1007/BF02773602

M3 - Article

AN - SCOPUS:33748570777

SN - 0021-2172

VL - 154

SP - 141

EP - 155

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

ER -