A model-theoretic analysis of Fidel-structures for mbC

In this paper the class of Fidel-structures for the paraconsistent logic mbC is studied from the point of view of Model Theory and Category Theory. The basic point is that Fidel-structures for mbC (or mbC-structures) can be seen as first-order structures over the signature of Boolean algebras expanded by two binary predicate symbols N (for negation) and O (for the consistency connective) satisfying certain Horn sentences. This perspective allows us to consider notions and results from Model Theory in order to analyze the class of mbC-structures. Thus, substructures, union of chains, direct products, direct limits, congruences and quotient structures can be analyzed under this perspective. In particular, a Birkhoff-like representation theorem for mbC-structures as subdirect poducts in terms of subdirectly irreducible mbC-structures is obtained by adapting a general result for first-order structures due to Caicedo. Moreover, a characterization of all the subdirectly irreducible mbC-structures is also given. An alternative decomposition theorem is obtained by using the notions of weak substructure and weak isomorphism considered by Fidel for Cn-structures

Groups with right-invariant multiorders

A Cayley object for a group G is a structure on which G acts regularly as a group of automorphisms. The main theorem asserts that a necessary and sufficient condition for the free abelian group G of rank m to have the generic n-tuple of linear orders as a Cayley object is that m>n. The background to this theorem is discussed. The proof uses Kronecker's Theorem on diophantine approximation.Comment: 9 page

The Gauss-Manin connection on the Hodge structures

Pour tout sch\'ema simplicial complexe $X_{\bullet}$ il existe une application canonique $\nabla:H^{\ast}(X_{\bullet})\longrightarrow \Omega^1_{{\mathbb C}/{\mathbb Q}}\otimes H^{\ast}(X_{\bullet})$, appel\'ee la connexion de Gau\ss-Manin. Nous montrons qu'il existe une unique connexion fonctorielle sur toute structure de Hodge-Tate mixte ayant certaines propri\'et\'es de la connexion de Gau\ss-Manin. Cette connexion n'est pas int\'egrable en g\'en\'eral, et alors son int\'egrabilit\'e est une condition non triviale pour qu'une structure de Hodge soit g\'eom\'etrique. Dans des cas particuliers, je donne des formules explicites pour la connexion de Gau\ss-Manin sur la cohomologie singuli\`ere des vari\'et\'es alg\'ebriques sur ${\mathbb C}$ dans les termes de la structure de Hodge

Special transformations in algebraically closed valued fields

We present two of the three major steps in the construction of motivic integration, that is, a homomorphism between Grothendieck semigroups that are associated with a first-order theory of algebraically closed valued fields, in the fundamental work of Hrushovski and Kazhdan. We limit our attention to a simple major subclass of V-minimal theories of the form ACVF_S(0, 0), that is, the theory of algebraically closed valued fields of pure characteristic $0$ expanded by a (VF, Gamma)-generated substructure S in the language L_RV. The main advantage of this subclass is the presence of syntax. It enables us to simplify the arguments with many different technical details while following the major steps of the Hrushovski-Kazhdan theory.Comment: This is the published version of a part of the notes on the Hrushovski-Kazhdan integration theory. To appear in the Annals of Pure and Applied Logi