arXiv subjects
Publications and source records attributed to Hugo Luiz Mariano.
In this work we explore some applications of the notions of Institution and Π-Institution in the setting of propositional logics and establish a precise categorial relation between these notions, i.e., we provide a pair of functors that establishes an adjunction between the categories Inst and Π-Inst.
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.
The present work presents some results about the categorial relation between logics and its categories of structures. A (propositional, finitary) logic is a pair given by a signature and Tarskian consequence relation on its formula algebra. The logics are the objects in our categories of logics; the morphisms are certain signature morphisms that are translations between logics (\cite{AFLM1},\cite{AFLM2},\cite{AFLM3} \cite{FC}). Morphisms between algebraizable logics (\cite{BP}) are translations that preserves algebraizing pairs (\cite{MaMe}): they can be completely encoded by certain functors defined on the quasi-variety canonically associated to the algebraizable logics. This kind of results will be useful in the development of a categorial approach to the representation theory of general logics (\cite{MaPi1}, \cite{MaPi2}, \cite{AJMP}).
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 $(Mod(T), \preceq)$, 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 $AEC$ 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.
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 $\{\neg, \vee\}$ and $\{\neg',\rightarrow'\}$) 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.
We establish the following model-theoretic characterization: profinite $L$-structures, the cofiltered limits of finite $L$-structures,are retracts of ultraproducts of finite $L$-structures. As a consequence, any elementary class of $L$-structures axiomatized by $L$-sentences of the form $\forall \vec{x} (ψ_{0}(\vec{x}) \ra ψ_{1}(\vec{x}))$, where $ψ_{0}(\vec{x}),ψ_{1}(\vec{x})$ are existencial-positives $L$-formulas, is closed under the formation of profinite objects in the category {\bf L-mod}, the category of structures suitable for the language $L$ and $L$-homomorphisms.