Model Theory and its Applications
Org: Bradd Hart (McMaster), Thomas Kucera (Manitoba) and Rahim Moosa (Waterloo)
- JOHN BALDWIN, University of Illinois at Chicago
Notions of Excellence
Excellence is the key property for studying categoricity in infinitary
logic. But the notion is only clearly defined and fully worked out
for atomic models of a first order theory (thereby coding complete
sentences of Lw1, w). We will consider some of the
possible ways of extending the definition and some examples.
- PAUL BANKSTON, Marquette University, Milwaukee, Wisconsin
Chainability and Unidimensionality from a Model-Theoretic Perspective
On the surface, the textbook definitions of chainability and
unidimensionality-in the sense of covering dimension-are quite
similar. In this talk we use model theory to explore the assertion
that this similarity is only skin deep. In the case of dimension,
there is a beautiful theorem of E. Hemmingsen that allows us to give
a first-order characterization in terms of the language of lattices.
We show that no such characterization is possible for chainability
by proving that if k is any infinite cardinal and
B is an open lattice base for a continuum, then
B is elementarily equivalent to an open lattice base for
a continuum X, of weight k, such that X has a three-set
open cover admitting no chain open refinement.
- ALF DOLICH, University of Illinois at Chicago, Chicago IL 60607
Independence relations, universality, and SOP4
In  (elaborated upon by Dzamonja and Shelah in ), Shelah
studies, for a fixed theory T, whether under an appropriate forcing
it is consistent that at certain cardinals l only few models
of T of cardinality l are needed to elementarily embed all
models of T of cardinality l. In  Shelah showed that for
any simple theory T as well as for various non-simple examples this
is the case and that this universality property fails for any theory
with the combinatorial property SOP4. I will discuss work towards
developing a uniform framework, based on the existence of independence
relations with several weak properties, intended to capture all
theories T with the desirable universality property alluded to
Mirna Dzamonja and Saharon Shelah,
On properties of theories which preclude the existence of
Ann. Pure Appl. Logic 139(2006), 280-302.
The universality spectrum: consistency for more classes.
In: Combinatorics, Bolyai Society Mathematical Studies, 1993,
Toward classifying unstable theories.
Ann. Pure Appl. Logic 80(1996), 229-255.
- ALINA DUCA, University of Manitoba, Winnipeg, MB, R3T 2N2
Description of the injective modules over a principal left
and right ideal domain
Over a principal left and right ideal domain R every injective
module is a direct sum of indecomposable injective modules. One
indecomposable injective is the injective envelope (divisible hull) of
the module RR and is isomorphic to the division algebra Q of
R. The other indecomposable injective modules are (up to
isomorphism) in a one-to-one correspondence with the prime elements of
the ring (up to similarity).
Motivated by a classic treatment of O. Ore, I take advantage of the
factorization theory in R and investigate the internal structure of
an indecomposable injective left module E ¹ Q. I describe its
"layered" structure in terms of its elementary socle series (soca (E) )a, a concept which was introduced by
I. Herzog as the elementary analogue of the socle series of a module,
where the minimality condition on the pp-definable subgroups is used.
Since E has the descending chain condition on pp-definable
subgroups, the elementary socle series exhausts E. A complete
characterization of ( soca(E) )a is
In addition, I will analyze the relationship between the classical
socle series of the right module E over the ring T = EndR(E)op and the elementary socle series of the left
- DRAGOS GHIOCA, McMaster University
A dynamical version of the Mordell-Lang Theorem
We prove a dynamical version of the Mordell-Lang conjecture for the
affine line. We use methods from number theory and model theory.
>From number theory we use analytic methods similar to the ones
employed by Skolem, Chabauty and Coleman for studying diophantine
equations. From model theory we use an uniform statement due to
Scanlon for the Manin-Mumford problem on the additive group scheme.
- BRADD HART, McMaster University
Group existence in simple continuous theories
This is a preliminary report on joint work with Jean-Martin Albert on
the existence of definable groups in simple, first-order continuous,
- DIERDRE HASKELL, McMaster University, 1280 Main St. W, Hamilton, ON, L8P 2T4
Stable domination and algebraically closed valued fields
The concept of stable domination has been developed as a way to extend
the tools of stability theory to structures which are not stable but
have a rich stable reduct (in a precise sense). In this talk, I will
define stable domination and its associated notion of domination
independence, and illustrate how some standard properties of
independence in a stable theory lift to corresponding properties of
stably dominated types. I will illustrate these properties with
examples in algebraically closed valued fields.
- IVO HERZOG, The Ohio State University, 4240 Campus Dr., Lima, OH 45804
Definable subspaces of finite dimensional representations
Let k be an algebraically closed field of characteristic 0, and
denote by L a finite-dimensional semisimple Lie algebra over k.
If f(v) is a positive-primitive formula in the language of
modules over the universal enveloping algebra U(L), then the subset
f(V) defined by f in an L-representation V is a subspace
of V, considered as a vector space over k. If f(V) defines a
sum of weight spaces of V, for every finite-dimensional
representation V, then there is a positive-primitive formula
f- that defines an orthogonal complement of f(V), for every
finite-dimensional V. The talk will be devoted to a proof of this
fact, as well as an explanation of the conjecture that if f(V)
defines the 1-simplex of weights whose boundary consists of the
highest weight space, and one of its conjugates under a simple
reflection, then f(V) must be a minimal linearly bounded
- THOMAS KUCERA, Department of Mathematics, University of Manitoba
Elementary Socles and Radicals
Socles and radicals are important tools in studying the structure of
modules and rings. The socle of a module is the sum of all of its
simple (minimal) submodules; dually the radical of a module is the
intersection of all of its maximal submodules. Ivo Herzog introduced
model-theoretic analogues of these concepts by replacing "submodule"
by "definable subgroup".
In an (indecomposable) totally transcendental module the elementary
socle is non-trivial and is a definably closed submodule.
Furthermore, the definition of elementary socle naturally extends to
an ascending series of definably closed submodules whose union is the
whole module. Dually, if an indecomposable pure-injective module has
the ascending chain condition on definable subgroups (ACC-pp), the
elementary radical is a proper submodule, and the definition of the
elementary radical may be extended to a descending series of
submodules whose intersection is 0.
Mike Prest introduced a notion of duality between certain first order
formulas in the languages of left modules and right modules which
Herzog extended to a duality of categories. This duality makes
indecomposable tt modules correspond to indecomposable pure-injective
modules with ACC-pp. I show that there is a natural similarity
between the structure of the elementary socle series of an
indecomposable tt module and the structure of the elementary radical
series of its elementary dual.
The elementary socle series has had limited application in describing
the structure of certain indecomposable injective modules; however
serious applications await a deeper understanding of the properties of
these series in general.
- SALMA KUHLMANN, University of Saskatchewan, 106 Wiggins Road, Saskatoon, SK
Restricted Exponentiation in Fields of Algebraic Power
Ron Brown (1971) proved that a valued vector space of countable
dimension admits a valuation basis. This result was applied by
S. Kuhlmann (2000) to show that every countable real closed field
admits a restricted exponential function, that is, an order preserving
isomorphism from the ideal of infinitesimals (MK, +) onto
the group of 1-units (1 + MK, ·). A natural question
arose whether every real closed field admits a restricted exponential
function. In this talk, we give a partial answer to this question.
To this end, we investigate valued fields which admit a valuation
basis (as valued vector spaces over a given countable ground field
K). We isolate a property (called TDRP in our paper) for a valued
subfield L of a field of generalized power series F((G)) (where
G is a countable ordered abelian group and F is a real closed, or
algebraically closed field). We show that this property implies the
existence of a K-valuation basis for L. In particular, we deduce
that the field of rational functions F(G) (the quotient field of the
group ring F[G]) and the field F(G) of power series in
F((G)) algebraic over F(G) admit K-valuation bases. If moreover
F is archimedean and G is divisible, we conclude that the real
closed field F(G) admits a restricted exponential function.
- CHRIS LASKOWSKI, University of Maryland
Ideals of formulas, 2-cardinal models, and group existence
We survey a host of results involving ideals of formulas, both inside
and outside the context of stability. In some cases we construct
2-cardinal models, while in others we assert the existence of
- RAHIM MOOSA, University of Waterloo, Waterloo, Ontario, N2L 3G1
A consequence of the canonical base property
A stable theory of finite rank is said to have the Canonical Base
Property (CBP) if for any stationary type p, the type of the
canonical base of p over a realisation of p is (almost) internal
to the collection of all non-modular minimal types. No examples of
the failure of CBP are known. Inspired by the model theory of compact
complex manifold, but working in an arbitrary complete stable theory
T of finite rank with the CBP, we give a geometric characterisation
of when a stationary type is internal to the collection of all
non-modular minimal types. Our characterisation is based on, and
partially recovers, Campana's "second algebraicity criterion" from
complex analytic geometry.
This is joint work with Anand Pillay.
- PHILIPP ROTHMALER, CUNY, University Ave & W 181 Street, Bronx, NY 10453
Cotorsion modules represent a homologically defined generalization of
pure-injective modules. There are cotorsion abelian groups that are
not pure-injective, but over von Neumann regular rings (that is, rings
over which all modules have quantifier elimination) and over pure
semisimple rings (that is, rings over which every module is totally
transcendental), this generalization yields nothing new. Joint work
with Ivo Herzog will be presented that characterizes the class of
rings (containing the former two classes) over which every cotorsion
module is pure-injective.
- THOMAS SCANLON, University of California Berkeley, Department of
Mathematics, Evans Hall, Berkeley, CA 94720-3840, USA
Trivial types and rational dynamics
Recall that a periodic point of a polynomial f(x) Î C[x]
is a complex number a for which fm(a) = a for some positive
integer m. We discuss how for certain choices of polynomials the
algebraic relations amongst the periodic points may be understood
through the study of associated trivial types in the theory of
- PATRICK SPEISSEGGER, McMaster University, 1280 Main St. W, Hamilton, ON, L8S 4K1
A reasonably tame Cantor set
We construct a Cantor set E Ì [0,1] such that for every n Î N and every bounded f: A® Rm definable in any
polynomially bounded o-minimal expansion of the real field, the image
f(En ÇA) is Minkowski null. It follows that the expansion of
the real field by E does not define the set of all natural numbers.
Joint work with Harvey Friedman, Krzysztof Kurdyka and Chris Miller.
- YEVGENIY VASILYEV, University of Windsor, Department of Mathematics and
Statistics, Windsor, ON, N9B 3P4
Externally definable sets in simple structures
I will talk about the issue of quantifier elimination in the expansion
of a simple structure with the traces of relations definable in some
elementary extension. By Shelah's result, in the case of theories
without the independence property, q.e. holds if we add all such
traces (with parameters from a saturated extension). Since q.e. fails if we apply the same procedure to a random graph, in the simple
case we restrict to parameters coming from a "lovely pair"
extension. In this setting, in a joint work with Anand Pillay, we
show that the expansion having q.e. is equivalent to a certain
definability condition (weak lowness) on the base theory, and find an
example of a non-weakly low simple theory. I will also discuss other
possible ways of adding externally definable sets, such as expansions
by "generic" traces.