61 research outputs found
Homomorphisms on infinite direct products of groups, rings and monoids
We study properties of a group, abelian group, ring, or monoid which (a)
guarantee that every homomorphism from an infinite direct product
of objects of the same sort onto factors through the direct product of
finitely many ultraproducts of the (possibly after composition with the
natural map or some variant), and/or (b) guarantee that when a
map does so factor (and the index set has reasonable cardinality), the
ultrafilters involved must be principal.
A number of open questions, and topics for further investigation, are noted.Comment: 26 pages. Copy at http://math.berkeley.edu/~gbergman/papers may be
updated more frequently than arXiv copy. Version 2 has minor revisions in
wording etc. from version
Model Theory for a Compact Cardinal
We would like to develop model theory for T, a complete theory in
L_{theta,theta}(tau) when theta is a compact cardinal. We already have bare
bones stability theory and it seemed we can go no further. Dealing with
ultrapowers (and ultraproducts) naturally we restrict ourselves to "D a
theta-complete ultrafilter on I, probably (I,theta)-regular". The basic
theorems of model theory work and can be generalized (like Los theorem), but
can we generalize deeper parts of model theory? The first section tries to sort
out what occurs to the notion of stable T for complete L_{theta,theta}-theories
T. We generalize several properties of complete first order T, equivalent to
being stable (see [Sh:c]) and find out which implications hold and which fail.
In particular, can we generalize stability enough to generalize [Sh:c, Ch. VI]?
Let us concentrate on saturation in the local sense (types consisting of
instances of one formula). We prove that at least we can characterize the T's
(of cardinality < theta for simplicity) which are minimal for appropriate
cardinal lambda > 2^kappa +|T| in each of the following two senses. One is
generalizing Keisler order which measures how saturated are ultrapowers.
Another asks: Is there an L_{theta,theta}-theory T_1 supseteq T of cardinality
|T| + 2^theta such that for every model M_1 of T_1 of cardinality > lambda, the
tau(T)-reduct M of M_1 is lambda^+-saturated. Moreover, the two versions of
stable used in the characterization are different
Constructing regular ultrafilters from a model-theoretic point of view
This paper contributes to the set-theoretic side of understanding Keisler's
order. We consider properties of ultrafilters which affect saturation of
unstable theories: the lower cofinality \lcf(\aleph_0, \de) of
modulo \de, saturation of the minimum unstable theory (the random graph),
flexibility, goodness, goodness for equality, and realization of symmetric
cuts. We work in ZFC except when noted, as several constructions appeal to
complete ultrafilters thus assume a measurable cardinal. The main results are
as follows. First, we investigate the strength of flexibility, detected by
non-low theories. Assuming is measurable, we construct a
regular ultrafilter on which is flexible (thus: ok) but
not good, and which moreover has large \lcf(\aleph_0) but does not even
saturate models of the random graph. We prove that there is a loss of
saturation in regular ultrapowers of unstable theories, and give a new proof
that there is a loss of saturation in ultrapowers of non-simple theories.
Finally, we investigate realization and omission of symmetric cuts, significant
both because of the maximality of the strict order property in Keisler's order,
and by recent work of the authors on . We prove that for any , assuming the existence of measurable cardinals below ,
there is a regular ultrafilter on such that any -ultrapower of
a model of linear order will have alternations of cuts, as defined below.
Moreover, will -saturate all stable theories but will not
-saturate any unstable theory, where is the smallest
measurable cardinal used in the construction.Comment: 31 page
Sobre los problemas de espectro de cofinalidad y p=t
La noción de problema de espectro de cofinalidad fue introducida en 2016 por Malliaris y Shelah en [11]. Esta noción permite conectar y dar respuesta a dos antiguos problemas abiertos en dos áreas totalmente distintas: el problema en Teoría de Modelos de determinar la maximalidad de SOP2 en el orden de Keisler y el problema en Topología Conjuntista de determinar si los cardinales invariantes del continuo p y t son iguales. En el presente trabajo hacemos un análisis detallado de la noción de problema de espectro de cofinalidad y su conexión con el problema de p = t. Además, estudiamos algunas aplicaciones topológicas de p = t y damos respuesta a una pregunta abierta hecha por TodorcevÍc y Velickovíc en [20] sobre la existencia de un conjunto parcialmente ordenado de tamaño p sin precalibre p como una consecuencia directa de p = tAbstract: The notion of cofinality spectrum problem was introduced by Malliaris and Shelah in [11]. This notion allows to connect and solve two longstanding open problems in quite different areas: the model-theoretic question of determining the maximality of SOP2-theories in Keisler’s order and the set-theoretic Topology problem of determining whether the cardinal invariants of the continuum p y t are the same. In the present dissertation we do a detailed analysis of the notion of cofinality spectrum problem and its connection with the problem p = t. Also, we study some topological applications of p = t and we answer an open question asked by Todorcevíc y Velickovíc in [20] about the existence of a poset of size p without precaliber p as a direct consequence of p = t.Maestrí
Sobre los problemas de espectro de cofinalidad y p=t
La noción de problema de espectro de cofinalidad fue introducida en 2016 por Malliaris y Shelah en [11]. Esta noción permite conectar y dar respuesta a dos antiguos problemas abiertos en dos áreas totalmente distintas: el problema en Teoría de Modelos de determinar la maximalidad de SOP2 en el orden de Keisler y el problema en Topología Conjuntista de determinar si los cardinales invariantes del continuo p y t son iguales. En el presente trabajo hacemos un análisis detallado de la noción de problema de espectro de cofinalidad y su conexión con el problema de p = t. Además, estudiamos algunas aplicaciones topológicas de p = t y damos respuesta a una pregunta abierta hecha por TodorcevÍc y Velickovíc en [20] sobre la existencia de un conjunto parcialmente ordenado de tamaño p sin precalibre p como una consecuencia directa de p = tAbstract: The notion of cofinality spectrum problem was introduced by Malliaris and Shelah in [11]. This notion allows to connect and solve two longstanding open problems in quite different areas: the model-theoretic question of determining the maximality of SOP2-theories in Keisler’s order and the set-theoretic Topology problem of determining whether the cardinal invariants of the continuum p y t are the same. In the present dissertation we do a detailed analysis of the notion of cofinality spectrum problem and its connection with the problem p = t. Also, we study some topological applications of p = t and we answer an open question asked by Todorcevíc y Velickovíc in [20] about the existence of a poset of size p without precaliber p as a direct consequence of p = t.Maestrí
- …