248 research outputs found
Quantifier elimination and other model-theoretic properties of BL-algebras
This work presents a model-theoretic approach to the study of firstorder theories of classes of BL-chains. Among other facts, we present several classes of BL-algebras, generating the whole variety of BL-algebras whose firstorder theory has quantifier elimination. Model-completeness and decision problems are also investigated. Then we investigate classes of BL-algebras having (or not having) the amalgamation property or the joint embedding property and we relate the above properties to the existence of ultrahomogeneous models. © 2011 by University of Notre Dame.Peer Reviewe
Quantifier elimination in quasianalytic structures via non-standard analysis
The paper is a continuation of our earlier article where we developed a
theory of active and non-active infinitesimals and intended to establish
quantifier elimination in quasianalytic structures. That article, however, did
not attain full generality, which refers to one of its results, namely the
theorem on an active infinitesimal, playing an essential role in our
non-standard analysis. The general case was covered in our subsequent preprint,
which constitutes a basis for the approach presented here. We also provide a
quasianalytic exposition of the results concerning rectilinearization of terms
and of definable functions from our earlier research. It will be used to
demonstrate a quasianalytic structure corresponding to a Denjoy-Carleman class
which, unlike the classical analytic structure, does not admit quantifier
elimination in the language of restricted quasianalytic functions augmented
merely by the reciprocal function. More precisely, we construct a plane
definable curve, which indicates both that the classical theorem by J. Denef
and L. van den Dries as well as \L{}ojasiewicz's theorem that every subanalytic
curve is semianalytic are no longer true for quasianalytic structures. Besides
rectilinearization of terms, our construction makes use of some theorems on
power substitution for Denjoy-Carleman classes and on non-extendability of
quasianalytic function germs. The last result relies on Grothendieck's
factorization and open mapping theorems for (LF)-spaces. Note finally that this
paper comprises our earlier preprints on the subject from May 2012.Comment: Final version, 36 pages. arXiv admin note: substantial text overlap
with arXiv:1310.130
Modular functionals and perturbations of Nakano spaces
We settle several questions regarding the model theory of Nakano spaces left
open by the PhD thesis of Pedro Poitevin \cite{Poitevin:PhD}. We start by
studying isometric Banach lattice embeddings of Nakano spaces, showing that in
dimension two and above such embeddings have a particularly simple and rigid
form. We use this to show show that in the Banach lattice language the modular
functional is definable and that complete theories of atomless Nakano spaces
are model complete. We also show that up to arbitrarily small perturbations of
the exponent Nakano spaces are -categorical and -stable. In
particular they are stable
The dynamical hierarchy for Roelcke precompact Polish groups
We study several distinguished function algebras on a Polish group , under
the assumption that is Roelcke precompact. We do this by means of the
model-theoretic translation initiated by Ben Yaacov and Tsankov: we investigate
the dynamics of -categorical metric structures under the action of
their automorphism group. We show that, in this context, every strongly
uniformly continuous function (in particular, every Asplund function) is weakly
almost periodic. We also point out the correspondence between tame functions
and NIP formulas, deducing that the isometry group of the Urysohn sphere is
\Tame\cap\UC-trivial.Comment: 25 page
Advances in the theory of μŁΠ algebras
Recently an expansion of ŁΠ1/2 logic with fixed points has been considered [23]. In the present work we study the algebraic semantics of this logic, namely μŁΠ algebras, from algebraic, model theoretic and computational standpoints. We provide a characterisation of free μŁΠ algebras as a family of particular functions from [0,1]n to [0,1]. We show that the first-order theory of linearly ordered μŁΠ algebras enjoys quantifier elimination, being, more precisely, the model completion of the theory of linearly ordered ŁΠ1/2 algebras. Furthermore, we give a functional representation of any ŁΠ1/2 algebra in the style of Di Nola Theorem for MV-algebras and finally we prove that the equational theory of μŁΠ algebras is in PSPACE. © The Author 2010. Published by Oxford University Press. All rights reserved.Marchioni acknowledges partial support of the Spanish projects MULOG2 (TIN2007-68005-C04), Agreement Technologies (CONSOLIDER CSD2007-0022, INGENIO 2010), the Generalitat de Catalunya grant 2009-SGR-1434, and Juan de la Cierva Program of the Spanish MICINN, as well as the ESF Eurocores-LogICCC/MICINN project (FFI2008-03126-E/FILO). Spada acknowledges partially supported of the FWF project P 19872-N18.Peer Reviewe
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