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.
Keywords
- Boolean Algebra
- Quotient Space
- Regular Cardinal
- Quotient Algebra
- Interval Space
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© 1984 Springer-Verlag
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Bonnet, R. (1984). On homomorphism types of superatomic interval Boolean algebras. In: Müller, G.H., Richter, M.M. (eds) Models and Sets. Lecture Notes in Mathematics, vol 1103. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0099381
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DOI: https://doi.org/10.1007/BFb0099381
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