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Saharon Shelah

Publications and source records attributed to Saharon Shelah.

At least 397 records · Page 22Linked to original sources
arXiv2007

Polish Algebras shy from freedom

Our first motivation was the question: can a countable structure have an automorphism group, which a free uncountable group? This is answered negatively in [Sh:744]. Lecturing in a conference in Rutgers, February 2001, I was asked whether I am really speaking on Polish groups. We can prove this using a more restrictive condition on the set of equations. Parallel theorems, hold for semi groups and for metric algebras, e.g. with non-isolated unit. Here we do the general case. For instance we show that there is no Polish group which as a group is free and uncountable.

math.LO↗
arXiv2007

All creatures great and small

Let kappa be an uncountable regular cardinal. Assuming 2^kappa=kappa^+, we show that the clone lattice on a set of size kappa is not dually atomic.

math.RA↗
arXiv2007

Noetherian ring with free additive groups

There are Noetherian rings (in fact domains) with a free additive group, in every infinite cardinality. (This is an expanded version of [SgSh:217] which appears in the Abstracts of the American Mathematical Society 7 (1986): 369.)

math.LO↗
arXiv2007

Abstract elementary classes near aleph_1

We prove in ZFC, no psi in L_{omega_1,omega}[Q] have unique model of uncountable cardinality, this confirms theBaldwin conjecture. But we analyze this in more general terms. We introduce and investigate a.e.c. and also versions of limit models, and prove some basic properties like representation by PC class, for any a.e.c. For PC_{aleph_0}-representable a.e.c. we investigate the conclusion of having not too many non-isomorphic models in aleph_1 and aleph_2, but have to assume 2^{aleph_0}<2^{aleph_1} and even 2^{aleph_1}<2^{aleph_2}.

math.LO↗
arXiv2007

The number of openly generated Boolean algebras

This article is devoted to two different generalizations of projective Boolean algebras: openly generated Boolean algebras and tightly sigma-filtered Boolean algebras. We show that for every uncountable regular cardinal kappa there are 2^kappa pairwise non-isomorphic openly generated Boolean algebras of size kappa > aleph_1 provided there is an almost free non-free abelian group of size kappa. The openly generated Boolean algebras constructed here are almost free. Moreover, for every infinite regular cardinal kappa we construct 2^kappa pairwise non-isomorphic Boolean algebras of size kappa that are tightly sigma-filtered and c.c.c. These two results contrast nicely with Koppelberg's theorem hat for every uncountable regular cardinal kappa there are only 2^{< kappa} isomorphism types of projective Boolean algebras of size kappa .

math.LO↗
arXiv2007

Winning the pressing down game but not Banach Mazur

Let $S$ be the set of those $α\inω_2$ that have cofinality $ω_1$. It is consistent relative to a measurable that the nonempty player wins the pressing down game of length $ω_1$, but not the Banach Mazur game of length $ω+1$ (both games starting with $S$).

math.LO↗
arXiv2007

Test Groups for Whitehead Groups

We consider the question of when the dual of a Whitehead group is a test group for Whitehead groups. This turns out to be equivalent to the question of when the tensor product of two Whitehead groups is Whitehead. We investigate what happens in different models of set theory.

math.LO↗
arXiv2007

On kernels of cellular covers

In the present paper we continue to examine cellular covers of groups, focusing on the cardinality and the structure of the kernel K of the cellular map G-> M . We show that in general a torsion free reduced abelian group M may have a proper class of non-isomorphic cellular covers. In other words, the cardinality of the kernels is unbounded. In the opposite direction we show that if the kernel of a cellular cover of any group M has certain ``freeness'' properties, then its cardinality must be bounded.

math.GR↗
arXiv2007

Lindelof spaces of singular density

A cardinal lambda is called omega-inaccessible if for all mu < lambda we have mu^omega<lambda. We show that for every omega-inaccessible cardinal lambda there is a CCC (hence cardinality and cofinality preserving) forcing that adds a hereditarily Lindelof regular space of density lambda. This extends an analogous earlier result of ours that only worked for regular lambda.

math.LO↗
arXiv2007

Resolvability vs. almost resolvability

A space X is kappa-resolvable (resp. almost kappa-resolvable) if it contains kappa dense sets that are pairwise disjoint (resp. almost disjoint over the ideal of nowhere dense subsets of X). Answering a problem raised by Juhasz, Soukup, and Szentmiklossy, and improving a consistency result of Comfort and Hu, we prove, in ZFC, that for every infinite cardinal {kappa} there is an almost 2^{kappa}-resolvable but not {omega}_1-resolvable space of dispersion character {kappa} .

math.GN↗
arXiv2007

On omitting cardinals in T_3 spaces

We prove that for every (infinite cardinal) lambda there is a T_3-space X with clopen basis, 2^mu points where mu = 2^lambda, such that every closed subspace of cardinality <|X| has cardinality < lambda .

math.LO↗
arXiv2006

Model theoretic stability and categoricity for complete metric spaces

We deal with the systematic development of stability for the context of approximate elementary submodels of a monster metric space, which is not far, but still very distinct from the first order case. In particular we prove the analogue of Morley's theorem for the classes of complete metric spaces.

math.LO↗
arXiv2006

Filtration equivalent aleph_1-separable abelian groups of cardinality aleph_1

We show that it is consistent with ordinary set theory ZFC and the generalized continuum hypothesis that there exist two separable abelian groups of cardinality aleph_1 which are filtration equivalent and one is a Whitehead group but the other is not. This solves one of the open problems of Eklof and Mekler.

math.LO↗
arXiv2006

Power set modulo small, the singular of uncountable cofinality

Let mu be singular of uncountable cofinality. If mu>2^{cf(mu)}, we prove that in P=([mu]^mu,supseteq) as a forcing notion we have a natural complete embedding of Levy(aleph_0, mu^+) (so P collapses mu^+ to aleph_0) and even Levy(aleph_0, U_{J^{bd}_kappa}(mu)) . The ``natural'' means that the forcing ({p in [mu]^mu :p closed}, supseteq) is naturally embedded and is equivalent to the Levy algebra. If mu <2^{cf(mu)} we have weaker results.

math.LO↗
arXiv2006

On Weak and Strong Interpolation in Algebraic Logics

We show that there is a restriction, or modification of the finite-variable fragments of First Order Logic in which a weak form of Craig's Interpolation Theorem holds, but a strong form of this theorem does not hold. Translating these results into Algebraic Logic we obtain a finitely axiomatizable subvariety of finite dimensional Representable Cylindric Algebras that has the Strong Amalgamation Property, but does not have the Superamalgamation Property. This settles a conjecture of Pigozzi

math.LO↗
arXiv2006

Existence of EF-equivalent Non Isomorphic Models

We prove the existence of pairs of models of the same cardinality lambda which are very equivalent according to EF games, but not isomorphic. We continue the paper math.LO/0404222, but we don't rely on it.

math.LO↗