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Saharon Shelah

Publications and source records attributed to Saharon Shelah.

At least 217 records · Page 12Linked to original sources
arXiv2018

Torsion-Free Abelian Groups are Consistently $a Δ^1_2$-complete

Let $\mbox{TFAG}$ be the theory of torsion-free abelian groups. We show that if there is no countable transitive model of $ZFC^- + κ(ω)$ exists, then $\mbox{TFAG}$ is $a Δ^1_2$-complete; in particular, this is consistent with $ZFC$. We define the $α$-ary Schröder- Bernstein property, and show that $\mbox{TFAG}$ fails the $α$-ary Schröder-Bernstein property for every $α< κ(ω)$. We leave open whether or not $\mbox{TFAG}$ can have the $κ(ω)$-ary Schröder-Bernstein property; if it did, then it would not be $a Δ^1_2$-complete, and hence not Borel complete.

math.LO↗
arXiv2018

On the class of flat stable theories

A new notion of independence relation is given and associated to it, the class of flat theories, a subclass of strong stable theories including the superstable ones is introduced. More precisely, after introducing this independence relation, flat theories are defined as an appropriate version of superstability. It is shown that in a flat theory every type has finite weight and therefore flat theories are strong. Furthermore, it is shown that under reasonable conditions any type is non-orthogonal to a regular one. Concerning groups in flat theories, it is shown that type-definable groups behave like superstable ones, since they satisfy the same chain condition on definable subgroups and also admit a normal series of definable subgroup with semi-regular quotients.

math.LO↗
arXiv2018

Deeply concatenable subgroups might never be free

We show that certain algebraic structures lack freeness in the absence of the axiom of choice. These include some subgroups of the Baer-Specker group $\mathbb{Z}^ω$ and the Hawaiian earring group. Applications to slenderness, completely metrizable topological groups, length functions and strongly bounded groups are also presented

math.GR↗
arXiv2018

Abstract elementary classes stable in $\aleph_0$

We study abstract elementary classes (AECs) that, in $\aleph_0$, 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 $\aleph_0$. More precisely, there is a superlimit model of cardinality $\aleph_0$ and the class generated by this superlimit has a type-full good $\aleph_0$-frame (a local notion of nonforking independence) and a superlimit model of cardinality $\aleph_1$. We also give a supersimplicity condition under which the locality hypothesis follows from the rest.

math.LO↗
arXiv2018

Local character of Kim-independence

We show that NSOP$_{1}$ theories are exactly the theories in which Kim-independence satisfies a form of local character. In particular, we show that if $T$ is NSOP$_{1}$, $M\models T$, and $p$ is a type over $M$, then the collection of elementary substructures of size $\left|T\right|$ over which $p$ does not Kim-fork is a club of $\left[M\right]^{\left|T\right|}$ and that this characterizes NSOP$_{1}$. We also present a new phenomenon we call dual local-character for Kim-independence in NSOP$_{1}$-theories.

math.LO↗
arXiv2018

On some variants of the club principle

We study some asymptotic variants of the club principle. Along the way, we construct some forcings and use them to separate several of these principles

math.LO↗
arXiv2018

Two results on cardinal invariants at uncountable cardinals

We prove two ZFC theorems about cardinal invariants above the continuum which are in sharp contrast to well-known facts about these same invariants at the continuum. It is shown that for an uncountable regular cardinal $κ$, $\mathfrak{b}(κ) = κ^{+}$ implies $\mathfrak{a}(κ) = κ^{+}$. This improves an earlier result of Blass, Hyttinen, and Zhang. It is also shown that if $κ\geq {\beth}_ω$ is an uncountable regular cardinal, then $\mathfrak{d}(κ) \leq \mathfrak{r}(κ)$. This result partially dualizes an earlier theorem of the authors.

math.LO↗
arXiv2018

Models of PA: Standard Systems without Minimal Ultrafilters

We prove that bold N, the standard model of arithmetic, has an uncountable elementary extension N such that there is no ultrafilter on the Boolean Algebra of subsets of bold N represented in N which is minimal (i.e. as in Rudin-Keisler order for partitions represented in N).

math.LO↗
arXiv2018

Polish Topologies for Graph Products of Cyclic Groups

We give a complete characterization of the graph products of cyclic groups admitting a Polish group topology, and show that they are all realizable as the group of automorphisms of a countable structure. In particular, we characterize the right-angled Coxeter groups (resp. Artin groups) admitting a Polish group topology. This generalizes results from [5], [7] and [4].

math.LO↗
arXiv2018

On the cofinality of the splitting number

The splitting number can be singular. The key method is to construct a forcing poset with finite support matrix iterations of ccc posets introduced by Blass and the second author "Ultrafilters with small generating sets", Israel J. Math., 65, (1989)

math.LO↗
arXiv2018

Asymmetric tie-points and almost clopen subsets of N*

A tie-point of a compact space is analogous to a cut-point: the complement of the point falls apart into two relatively clopen non-compact subsets. Set-theoretically a tie-point of N* is an ultrafilter whose dual maximal ideal can be generated by the union of two non-principal mod finite orthogonal ideals. We review some of the many consistency results that have depended on the construction of tie-points of N*. One especially important application, due to Velickovic, was to the existence of non-trivial involutions on N*. A tie-point of N* has been called symmetric if it is the unique fixed point of an involution. We define the notion of an almost clopen set to be the closure of one of the proper relatively clopen subsets of the complement of a tie-point. We explore asymmetries of almost clopen subsets of N* in the sense of how may an almost clopen set differ from its natural complementary almost clopen set.

math.GN↗
arXiv2017

On cuts in ultraproducts of linear orders II

We continue our study of the class $\mathscr{C}(D)$, where $D$ is a uniform ultrafilter on a cardinal $κ$ and $\mathscr{C}(D)$ is the class of all pairs $(θ_1, θ_2),$ where $(θ_1, θ_2)$ is the cofinality of a cut in $J^κ/D$ and $J$ is some $(θ_1+θ_2)^+$-saturated dense linear order. We give a combinatorial characterization of the class $\mathscr{C}(D)$. We also show that if $(θ_1, θ_2) \in \mathscr{C}(D)$ and $D$ is $\aleph_1$-complete or $θ_1 + θ_2 > 2^κ,$ then $θ_1=θ_2.$

math.LO↗
arXiv2017

An extendible model with a rigid elementary extension

A countable structure is said to be extendible if it has the same Scott sentence as some uncountable structure. Rigid structures are not extendible. We give an example of an extendible model with a rigid elementary extension.

math.LO↗
arXiv2017

On $C^s_n(κ)$ and the Juhasz-Kunen question

We generalize the combinatorial principles $C_n(κ), C^s_n(κ)$ and $Princ(κ)$ introduced by various authors, and prove some of their properties and connections between them. We also answer a question asked by Juhasz-Kunen about the relation between these principles, by showing that $C_n(κ)$ does not imply $C_{n+1}(κ)$, for any $n>2$. We also show the consistency of $C(κ)+\neg C^s(κ).$

math.LO↗
arXiv2017

The abelianization of inverse limits of groups

The abelianization is a functor from groups to abelian groups, which is left adjoint to the inclusion functor. Being a left adjoint, the abelianization functor commutes with all small colimits. In this paper we investigate the relation between the abelianization of a limit of groups and the limit of their abelianizations. We show that if $\mathcal{T}$ is a countable directed poset and $G:\mathcal{T}\longrightarrow\mathcal{G} rp$ is a diagram of groups that satisfies the Mittag-Leffler condition, then the natural map $$\mathrm{Ab}(\lim_{t\in\mathcal{T}}G_t)\longrightarrow\lim_{t\in\mathcal{T}}\mathrm{Ab}(G_t)$$ is surjective, and its kernel is a cotorsion group. In the special case of a countable product of groups, we show that the Ulm length of the kernel does not exceed $\aleph_1$.

math.GR↗
arXiv2017

Group Metrics for Graph Products of Cyclic Groups

We complement the characterization of the graph products of cyclic groups $G(Γ, \mathfrak{p})$ admitting a Polish group topology of [9] with the following result. Let $G = G(Γ, \mathfrak{p})$, then the following are equivalent: (i) there is a metric on $Γ$ which induces a separable topology in which $E_Γ$ is closed; (ii) $G(Γ, \mathfrak{p})$ is embeddable into a Polish group; (iii) $G(Γ, \mathfrak{p})$ is embeddable into a non-Archimedean Polish group. We also construct left-invariant separable group ultrametrics for $G = G(Γ, \mathfrak{p})$ and $Γ$ a closed graph on the Baire space, which is of independent interest.

math.LO↗