226 research outputs found
Mutually algebraic structures and expansions by predicates
We introduce the notions of a mutually algebraic structures and theories and
prove many equivalents. A theory is mutually algebraic if and only if it is
weakly minimal and trivial if and only if no model of has an expansion
by a unary predicate with the finite cover property. We show that every
structure has a maximal mutually algebraic reduct, and give a strong structure
theorem for the class of elementary extensions of a fixed mutually algebraic
structure.Comment: Incorporated comments and suggestions of the anonymous referee. 16
page
Model companion of ordered theories with an automorphism
Kikyo and Shelah showed that if is a theory with the Strict Order
Property in some first-order language , then in the expanded
language with a new unary
function symbol , the bigger theory T_\sigma := T\cup\{``\sigma
\mbox{is an} \mathcal{L}\mbox{-automorphism''}\} does not have a model
companion. We show in this paper that if, however, we restrict the automorphism
and consider the theory as the base theory together with a
``restricted'' class of automorphisms, then can have a model
companion in . We show this in the context of linear orders
and ordered abelian groups
On VC-minimal theories and variants
In this paper, we study VC-minimal theories and explore related concepts. We
first define the notion of convex orderablility and show that this lies
strictly between VC-minimality and dp-minimality. Next, we define the notion of
weak VC-minimality, show it lies strictly between VC-minimality and dependence,
and show that all unstable weakly VC-minimal theories interpret an infinite
linear order. Finally, we define the notion full VC-minimality, show that this
lies strictly between weak o-minimality and VC-minimality, and show that
theories that are fully VC-minimal have low VC-density.Comment: 15 page
Weakly minimal groups with a new predicate
Fix a weakly minimal (i.e., superstable -rank ) structure
. Let be an expansion by constants for an
elementary substructure, and let be an arbitrary subset of the universe
. We show that all formulas in the expansion are
equivalent to bounded formulas, and so is stable (or NIP) if
and only if the -induced structure on is
stable (or NIP). We then restrict to the case that is a pure
abelian group with a weakly minimal theory, and is mutually
algebraic (equivalently, weakly minimal with trivial forking). This setting
encompasses most of the recent research on stable expansions of
. Using various characterizations of mutual algebraicity, we
give new examples of stable structures of the form . Most
notably, we show that if is a weakly minimal additive subgroup of the
algebraic numbers, is enumerated by a homogeneous linear
recurrence relation with algebraic coefficients, and no repeated root of the
characteristic polynomial of is a root of unity, then is
superstable for any .Comment: 23 pages, final version incorporating referee comment
Karp height of models of stable theories
A trichotomy theorem for countable, stable, unsuperstable theories is
offered. We develop the notion of a `regular ideal' of formulas and study types
that are minimal with respect to such an ideal
Forcing Isomorphism II
If T has only countably many complete types, yet has a type of infinite
multiplicity then there is a ccc forcing notion Q such that, in any Q --generic
extension of the universe, there are non-isomorphic models M_1 and M_2 of T
that can be forced isomorphic by a ccc forcing. We give examples showing that
the hypothesis on the number of complete types is necessary and what happens if
`ccc' is replaced other cardinal-preserving adjectives. We also give an example
showing that membership in a pseudo-elementary class can be altered by very
simple cardinal-preserving forcings
P-NDOP and P-decompositions of aleph_epsilon-saturated models of superstable theories
Assume a complete superstable theory is superstable, and let P be a class of
regular types, typically closed under automorphisms of the monster and
non-orthogonality. We define the notion of P-NDOP and prove the existence of
P-decompositions and derive an analog of Sh401 for superstable theories with
P-NDOP. In this context, we also find a sufficient condition on
P-decompositions that imply non-isomorphic models. For this, we investigate
natural structures on the types in P\intersect S(M) modulo non-orthogonality.Comment: [LwSh:933] Version 4 corrects typos from Version
Karp complexity and classes with the independence property
A class K of structures is controlled if for all cardinals lambda, the
relation of L_{infty,lambda}-equivalence partitions K into a set of equivalence
classes (as opposed to a proper class). We prove that no pseudo-elementary
class with the independence property is controlled. By contrast, there is a
pseudo-elementary class with the strict order property that is controlled
The Karp complexity of unstable classes
A class K of structures is controlled if, for all cardinals lambda, the
relation of L_{infty,lambda}-equivalence partitions K into a set of equivalence
classes (as opposed to a proper class). We prove that the class of doubly
transitive linear orders is controlled, while any pseudo-elementary class with
the omega-independence property is not controlled
On the existence of atomic models
We give an example of a countable theory T such that for every cardinal
lambda >= aleph_2 there is a fully indiscernible set A of power lambda such
that the principal types are dense over A, yet there is no atomic model of T
over A. In particular, T(A) is a theory of size lambda where the principal
types are dense, yet T(A) has no atomic model
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