20,710 research outputs found
On universal modules with pure embeddings
We show that certain classes of modules have universal models with respect to
pure embeddings.
Let be a ring, a first-order theory with an infinite model
extending the theory of -modules and (where
stands for pure submodule). Assume has joint embedding and
amalgamation.
If or ,
then has a universal model of cardinality .
As a special case we get a recent result of Shelah [Sh17, 1.2] concerning the
existence of universal reduced torsion-free abelian groups with respect to pure
embeddings.
We begin the study of limit models for classes of -modules with joint
embedding and amalgamation. We show that limit models with chains of long
cofinality are pure-injective and we characterize limit models with chains of
countable cofinality. This can be used to answer Question 4.25 of [Maz].
As this paper is aimed at model theorists and algebraists an effort was made
to provide the background for both.Comment: 17 page
Hanf Numbers and Presentation Theorems in AECs
We prove that a strongly compact cardinal is an upper bound for a Hanf number
for amalgamation, etc. in AECs using both semantic and syntactic methods. To
syntactically prove non-disjoint amalgamation, a different presentation theorem
than Shelah's is needed. This relational presentation theorem has the added
advantage of being {\it functorial}, which allows the transfer of amalgamation
Forking and superstability in tame AECs
We prove that any tame abstract elementary class categorical in a suitable
cardinal has an eventually global good frame: a forking-like notion defined on
all types of single elements. This gives the first known general construction
of a good frame in ZFC. We show that we already obtain a well-behaved
independence relation assuming only a superstability-like hypothesis instead of
categoricity. These methods are applied to obtain an upward stability transfer
theorem from categoricity and tameness, as well as new conditions for
uniqueness of limit models.Comment: 33 page
The categoricity spectrum of large abstract elementary classes
The categoricity spectrum of a class of structures is the collection of
cardinals in which the class has a single model up to isomorphism. Assuming
that cardinal exponentiation is injective (a weakening of the generalized
continuum hypothesis, GCH), we give a complete list of the possible
categoricity spectrums of an abstract elementary class with amalgamation and
arbitrarily large models. Specifically, the categoricity spectrum is either
empty, an end segment starting below the Hanf number, or a closed interval
consisting of finite successors of the L\"owenheim-Skolem-Tarski number (there
are examples of each type). We also prove (assuming a strengthening of the GCH)
that the categoricity spectrum of an abstract elementary class with no maximal
models is either bounded or contains an end segment. This answers several
longstanding questions around Shelah's categoricity conjecture.Comment: 50 page
A model theoretic solution to a problem of L\'{a}szl\'{o} Fuchs
Problem 5.1 in page 181 of [Fuc15] asks to find the cardinals such
that there is a universal abelian -group for purity of cardinality
, i.e., an abelian -group of cardinality such
that every abelian -group of cardinality purely embeds in
. In this paper we use ideas from the theory of abstract elementary
classes to show:
Let be a prime number. If
or , then there is a universal abelian -group for purity of
cardinality . Moreover for , there is a universal abelian
-group for purity of cardinality if and only if .
As the theory of abstract elementary classes has barely been used to tackle
algebraic questions, an effort was made to introduce this theory from an
algebraic perspective.Comment: 10 page
Symmetry and the Union of Saturated Models in Superstable Abstract Elementary Classes
Our main result (Theorem 1) suggests a possible dividing line
(-superstable -symmetric) for abstract elementary classes without
using extra set-theoretic assumptions or tameness. This theorem illuminates the
structural side of such a dividing line.
Theoerem 1: Let be an abstract elementary class with no maximal
models of cardinality which satisfies the joint embedding and
amalgamation properties. Suppose . If is
- and -superstable and satisfies -symmetry, then for any
increasing sequence of -saturated models,
is -saturated.
We also apply results of VanDieren's Superstability and Symmetry paper and
use towers to transfer symmetry from down to in abstract
elementary classes which are both - and -superstable:
Theorem 2: Suppose is an abstract elementary class satisfying
the amalgamation and joint embedding properties and that is both
- and -superstable. If has symmetry for
non--splitting, then has symmetry for non--splitting.Comment: This paper is a synthesis of arXiv:1507.01991 and arXiv:1507.0198
A survey on tame abstract elementary classes
Tame abstract elementary classes are a broad nonelementary framework for
model theory that encompasses several examples of interest. In recent years,
progress toward developing a classification theory for them have been made.
Abstract independence relations such as Shelah's good frames have been found to
be key objects. Several new categoricity transfers have been obtained. We
survey these developments using the following result (due to the second author)
as our guiding thread:
If a universal class is categorical in cardinals of arbitrarily high
cofinality, then it is categorical on a tail of cardinals.Comment: 84 page
Joint diamonds and Laver diamonds
The concept of jointness for guessing principles, specifically
and various Laver diamonds, is introduced. A family of
guessing sequences is joint if the elements of any given sequence of targets
may be simultaneously guessed by the members of the family. While equivalent in
the case of , joint Laver diamonds are nontrivial new
objects. We give equiconsistency results for most of the large cardinals under
consideration and prove sharp separations between joint Laver diamonds of
different lengths in the case of -supercompact cardinals.Comment: 34 pages; revised version with several improvements, including
expanded Sections 3.3 and
Tameness, Uniqueness and amalgamation
We combine two approaches to the study of classification theory of AECs: 1.
that of Shelah: studying non-forking frames without assuming the amalgamation
property but assuming the existence of uniqueness triples and 2. that of
Grossberg and VanDieren: (studying non-splitting) assuming the amalgamation
property and tameness. In [JrSh875], we derive a good non-forking
-frame from a semi-good non-forking -frame. But the classes
and are replaced:
is restricted to the saturated models and the partial order
is restricted to the partial order
. Here, we avoid the restriction of the partial order
, assuming that every saturated model (in
over ) is an amalgamation base and
-tameness for non-forking types over saturated models, (in
addition to the hypotheses of [JrSh875]): We prove that if and
only if , provided that and are
saturated models. We present sufficient conditions for three good non-forking
-frames: one relates to all the models of cardinality
and the two others relate to the saturated models only. By an `unproven claim'
of Shelah, if we can repeat this procedure times, namely, `derive'
good non-forking frame for each then the categoricity
conjecture holds. Vasey applies one of our main theorems in a proof of the
categoricity conjecture under the above `unproven claim' of Shelah and more
assumptions. In [Jrprime], we apply the main theorem in a proof of the
existence of primeness triples
Good Frames With A Weak Stability
Let K be an abstract elementary class of models. Assume that there are less
than the maximal number of models in K_{\lambda^{+n}} (namely models in K of
power \lambda^{+n}) for all n. We provide conditions on K_\lambda, that imply
the existence of a model in K_{\lambda^{+n}} for all n. We do this by providing
sufficiently strong conditions on K_\lambda, that they are inherited by a
properly chosen subclass of K_{\lambda^+}
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