On homomorphism types of superatomic interval Boolean algebras

Abstract

Let D_κ 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 B_α,m,n be the superatomic interval algebra generated by the chain ω^α · m + (ω^α*). n. Let ℵ_κ be the subset of D_κ consisting of all B_α,m,n. In the first part, we consider the following relation in D_κ : B′ ⩽ B″ iff B′ is embeddable in B″. We prove that for every B in D_κ, there is an unique B_α,m,n such that B ⩽ B_α,m,n ⩽ B. We describe completely 〈ℵ_κ, ⩽ 〉 : this is a well-founded distributive lattice with the property that for every B_α,m,n, there are only finitely many incomparable elements to B_α,m,n in ℵ_κ. In the second part, we introduce other quasi-orderings ≦ on D_κ : 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 〈D_κ, ⩽ 〉 is very complicated : to each subset I of κ, we can associate a member B_I of B_κ, such that I ⊂ J if B_I ≦ B_J.

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.