Jonsson algebras in successor cardinals

Israel Journal of Mathematics - We shall show here that in many successor cardinals λ, there is a Jonsson algebra (in other words Jn(λ), or λ is not a Jonsson cardinal). In...     

Separability properties of almost — disjoint families of sets

We solve here some problems arising from a work by Hechler [3]. We eliminate extra set-theoretic axioms (MA, in fact) from existence theorems and deal with the existence of disjoint sets.     

On nice equivalence relations on <Superscript>λ</Superscript>2

Let E be an equivalence relation on the powerset of an uncountable set, which is reasonably definable. We assume that any two subsets with symmetric difference of size exactly 1 are not equivalent. We investigate whether for E there are many pairwise non equivalent sets. I would like to thank Alice Leonhardt for the beautiful typing.This research was supported by The Israel Science Foundation. Publication 724. Mathematics Subject Classification (2000): 03E47, 03E35; 20K20, 20K35     

On needed reals

Given a binary relationR, we call a subsetA of the range ofR R-adequate if for everyx in the domain there is someyεA such that (x, y)εR. Following Blass [4], we call a realη ”needed” forR if in everyR-adequate set we find an element from whichη is Turing computable. We show that every real needed for inclusion on the Lebesgue null sets,Cof(\(\mathcal{N}\)), is hyperarithmetic. Replacing “R-adequate” by “R-adequate with minimal cardinality” we get the related notion of being “weakly needed”. We show that it is consistent that the two notions do not coincide for the reaping relation. (They coincide in many models.) We show that not all hyperarithmetic reals are needed for the reaping relation. This answers some questions asked by Blass at the Oberwolfach conference in December 1999 and in [4].     

Functorial equations for lexicographic products

We generalize the main result of an earlier paper by the authors (Exponentiation in power series fields, Proc. Amer. Math. Soc. 125 (1997), 3177-3183) concerning the convex embeddings of a chain in a lexicographic power . For a fixed non-empty chain , we derive necessary and sufficient conditions for the existence of non-empty solutions to each of the lexicographic functorial equations

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A partition relation using strongly compact cardinals

If If is strongly compact and \kappa $"> and is regular (or alternatively cf , then holds for .

     

Gently killing S-spaces

We produce a model of ZFC in which there are no locally compact first countable S-spaces, and in which 2^ℵ ₀<2^ℵ ₁. A consequence of this is that in this model there are no locally compact, separable, hereditarily normal spaces of size ℵ₁, answering a question of the second author [9]. The first author would like to thank The Hebrew University of Jerusalem for their support while the research in this paper was being carried out. The research of the second author was partially supported by NSF Grant DMS-9322613. The research of the third author was partially supported by NSF grant DMS-9704477 and the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities. This is publication number 690 in the list of the third author.     

A new look at interpretability and saturation

We investigate the interpretability ordering ${?}^{?}$ using generalized Ehrenfeucht–Mostowski models. This gives a new approach to proving inequalities and investigating the structure of types.     

Abstract elementary classes stable in ℵ0

We study abstract elementary classes (AECs) that, in ${\mathrm{?}}_0$, have amalgamation, joint embedding, no maximal models and are stable (in terms of the number of orbital types). Assuming a locality property for types, we prove that such classes exhibit superstable-like behavior at ${\mathrm{?}}_0$. More precisely, there is a superlimit model of cardinality ${\mathrm{?}}_0$ and the class generated by this superlimit has a type-full good ${\mathrm{?}}_0$-frame (a local notion of nonforking independence) and a superlimit model of cardinality ${\mathrm{?}}_1$. We also give a supersimplicity condition under which the locality hypothesis follows from the rest.