On homomorphism types of superatomic interval Boolean algebras

Abstract

Let Dκ be the class of superatomic interval Boolean algebras of cardinality κ ⩾ ω1. For κ ⩽ α < κ+ and for m,n < ω, m+n ⩾ 1, let Bα,m,n be the superatomic interval algebra generated by the chain ωα · m + (ωα*). n. Let ℵκ be the subset of Dκ consisting of all Bα,m,n. In the first part, we consider the following relation in Dκ : B′ ⩽ B″ iff B′ is embeddable in B″. We prove that for every B in Dκ, there is an unique Bα,m,n such that B ⩽ Bα,m,n ⩽ B. We describe completely 〈ℵκ, ⩽ 〉 : this is a well-founded distributive lattice with the property that for every Bα,m,n, there are only finitely many incomparable elements to Bα,m,n in ℵκ. In the second part, we introduce other quasi-orderings ≦ on Dκ : for instance the relations being elementary embeddable, being a homomorphic image, being a dense homomorphic image. In contrast to the first part, for these relations ≦, the quasi-ordered class 〈Dκ, ⩽ 〉 is very complicated : to each subset I of κ, we can associate a member BI of Bκ, such that I ⊂ J if BI ≦ BJ.We thank the referees, I.ROSENBERG and S.KOPPELBERG for their comments, in particular concerning the proof of the theorem in § I, and S.SHELAH for his helpful comments and improvements of results in § II.KeywordsBoolean AlgebraQuotient SpaceRegular CardinalQuotient AlgebraInterval SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.