642 research outputs found
Profinite Structures are Retracts of Ultraproducts of Finite Structures
We establish the following model-theoretic characterization: profinite
-structures, the cofiltered limits of finite -structures,are retracts of
ultraproducts of finite -structures. As a consequence, any elementary class
of -structures axiomatized by -sentences of the form \forall \vec{x}
(\psi_{0}(\vec{x}) \ra \psi_{1}(\vec{x})), where
are existencial-positives -formulas,
is closed under the formation of profinite objects in the category {\bf L-mod},
the category of structures suitable for the language and -homomorphisms
Some consequences of the Firoozbakht's conjecture
In this paper we present the statement of the Firoozbakht's conjecture, some
of its consequences if it is proved and we show a consequence of Zhang's
theorem concerning the Firoozbakht's conjecture
A Universal Algebraic Survey of Rings
In this paper we present some basic results of the Universal Algebra of
-rings which were nowhere to be found in the current
literature. The outstanding book of I. Moerdijk and G. Reyes,[24], presents the
basic (and advanced) facts about -rings, however such a
presentation has no universal algebraic "flavour". We have been inspired to
describe -rings through this viewpoint by D. Joyce in [15].
Our main goal here is to provide a comprehensive material with detailed proofs
of many known "taken for granted" results and constructions used in the
literature about -rings and their applications - such
proofs either could not be found or were merely sketched. We present, in
detail, the main constructions one can perform within this category, such as
limits, products, homomorphic images, quotients, directed colimits, free
objects and others, providing a "propaedeutic exposition" for the reader's
benefit
Representation theory of logics: a categorial approach
The major concern in the study of categories of logics is to describe
condition for preservation, under the a method of combination of logics, of
meta-logical properties. Our complementary approach to this field is study the
"global" aspects of categories of logics in the vein of the categories \Ss,
\Ls, \cA_s studied in \cite{AFLM3}. All these categories have good properties
however the category of logics \cL does not allow a good treatment of the
"identity problem" for logics (\cite{Bez}): for instance, the presentations of
"classical logics" (e.g., in the signature and
) are not \Ls-isomorphic. In this work, we sketch a
possible way to overcome this "defect" (and anothers) by a mathematical device:
a representation theory of logics obtained from category theoretic aspects on
(Blok-Pigozzi) algebraizable logics. In this setting we propose the study of
(left and right) "Morita equivalence" of logics and variants. We introduce the
concepts of logics (left/right)-(stably) -Morita-equivalent and show that the
presentations of classical logics are stably Morita equivalent but classical
logics and intuitionist logics are not stably-Morita-equivalent: they are only
stably-Morita-adjointly related.Comment: 10 page
Von Neumann Regular Rings and Applications
In this paper we present the notion of a von Neumann regular
ring, we prove some results about them and we describe
some of their properties. We prove, using two different methods, that the
category of von Neumann regular rings is a reflective
subcategory of . We prove that every
homomorphism between Boolean algebras can be represented by a
ring homomorphism between von Neumann regular
rings.Comment: 72 page
Topics on Smooth Commutative Algebra
We present, in the same vein as in [20] and [21], some results of the
so-called "Smooth (or ) Commutative Algebra", a version of
Commutative Algebra of rings instead of ordinary
commutative unital rings, looking for similar results to those one finds in the
latter, and expanding some others presented in [20]. We give an explicit
description of an adjunction between the categories and , in order to study this "bridge". We present and prove
many properties of the analog of the radical of an ideal of a ring (namely, the
-radical of an ideal), saturation (which we define as
"smooth saturation", inspired by [13]), rings of fractions
(-rings of fractions, defined first by I. Moerdijk and G.
Reyes in [20]), local rings (local -rings), reduced rings
(-reduced -rings) and others. We also
state and prove new results, such as ad hoc "Separation Theorems", similar to
the ones we find in Commutative Algebra, and a stronger version (Theorem 6) of
the Theorem 1.4 of [20], characterizing every -ring of
fractions. We describe the fundamental concepts of Order Theory for
-rings, proving that every -ring is
semi-real, and we prove an important result on the strong interplay between the
smooth Zariski spectrum and the real smooth spectrum of a
-ring
Towards a good notion of categories of logics
We consider (finitary, propositional) logics through the original use of
Category Theory: the study of the "sociology of mathematical objects", aligning
us with a recent, and growing, trend of study logics through its relations with
other logics (e.g. process of combinations of logics as bring [Gab] and
possible translation semantics [Car]). So will be objects of study the classes
of logics, i.e. categories whose objects are logical systems (i.e., a signature
with a Tarskian consequence relation) and the morphisms are related to (some
concept of) translations between these systems. The present work provides the
first steps of a project of considering categories of logical systems
satisfying simultaneously certain natural requirements: it seems that in the
literature ([AFLM1], [AFLM2], [AFLM3], [BC], [BCC1], [BCC2], [CG], [FC]) this
is achieved only partially.Comment: 16 page
A global approach to AECs
In this work we present some general categorial ideas on Abstract Elementary
Classes (AECs) %\cite{She}, inspired by the totality of AECs of the form
, for a first-order theory T: (i) we define a natural notion
of (funtorial) morphism between AECs; (ii) explore the following constructions
of AECs: "generalized" theories, pullbacks of AECs, (Galois) types as AECs;
(iii) apply categorial and topological ideas to encode model-theoretic notions
on spaces of types %(see Michael Lieberman Phd thesis) ; (iv) present the
"local" axiom for AECs here called "local Robinson's property" and an
application (Robinson's diagram method); (v) introduce the category of
Grothendieck's gluings of all AECs (with change of basis); (vi) introduce the
"global" axioms of "tranversal Robinson's property" (TRP) and "global
Robinson's property" (GRP) and prove that TRP is equivalent to GRP and GRP
entails a natural version of Craig interpolation property.Comment: 10 page
First-order characterization of noncommutative birational equivalence
Let be a root system with Weyl group . Let be an
algebraically closed field of zero characteristic, and consider the
corresponding semisimple Lie algebra . Then
there is a first-order sentence in the language
of rings sucht that, for any algebraically closed field
of , the validity of the Gelfand-Kirillov Conjecture for
is equivalent to . By the same method, we can show that the validity of
Noncommutative Noether's Problem for , any
algebraically closed field of is equivalent to , a formula in the same language. As consequences, we obtain
results on the modular Gelfand-Kirillov Conjecture and we show that, for
algebraically closed with characteristic ,
is a case of positive solution of modular Noncommutative
Noether's Problem.Comment: 12 page
On categories of o-minimal structures
Our aim in this paper is to look at some transfer results in model theory
(mainly in the context of o-minimal structures) from the category theory
viewpoint.Comment: 9 pages, 4 figure
- …