arXiv subjects
Publications and source records attributed to Hugo Luiz Mariano.
Forcing was first introduced by Paul J. Cohen in his work on the independence of the Continuum Hypothesis. Other formulations of forcing appeared using Model Theory, Boolean-valued Models, and Topos Theory. There is a folkloric claim that these three approaches are equivalent, at least at the level of their mathematical content. In this work, we present some results not found in the literature toward establishing connections between these versions of forcing.
Let $Σ$ be a (reduced) root system. Let $\mathsf{k}$ be an algebraically closed field of zero characteristic, and consider the corresponding semisimple Lie algebra $\mathfrak{g}_{\mathsf{k}, Σ}$. Then there is a first-order sentence $ϕ_Σ$ in the language $\mathcal{L}=(1,0,+,*,-)$ of rings such that, for any algebraically closed field $\mathsf{k}$ of characteristic 0, the validity of the Gelfand-Kirillov Conjecture for $\mathfrak{g}_{\mathsf{k}, Σ}$ is equivalent to $ACF_0 \vdash ϕ_Σ$.
We expand Čech cohomology of a topological space $X$ with values in a presheaf on $X$ to Čech cohomology of a commutative ring with unity $R$ with values in a presheaf on $R$. The strategy is to observe that both the set of open subsets of $X$ and the set of ideals of $R$ provide examples of a (semicartesian) quantale. We study a particular pair of (adjoint) functors $(θ, τ)$ between the quantale of open subsets of $X$ and the quantale of ideals of $C(X)$, the ring of real-valued continuous functions on $X$. This leads to the main result of this paper: the $q$th Čech cohomology groups of $X$ with values on the constant sheaf $F$ on $X$ is isomorphic to the $q$th Čech cohomology groups of the ring $C(X)$ with values on a sheaf $F \circ τ$ on $C(X)$.
In this paper, we present a generalization of Grothendieck pretopologies -- suited for semicartesian categories with equalizers $C$ -- leading to a closed monoidal category of sheaves, instead of closed cartesian category. This is proved through a different sheafification process, which is the left adjoint functor of the suitable inclusion functor but does not preserve all finite limits. If the monoidal structure in $C$ is given by the categorical product, all constructions coincide with those for Grothendieck toposes. The motivation for such generalization stems from a certain notion of sheaves on quantales that does not form a topos.
In this paper, we introduce the concept of a "von Neumann regular $\mathcal{C}^{\infty}$-ring", which is a model for a specific equational theory. We delve into the characteristics of these rings and demonstrate that each Boolean space can be effectively represented as the image of a von Neumann regular $\mathcal{C}^{\infty}$-ring through a specific functor. Additionally, we establish that every homomorphism between Boolean algebras can be expressed through a $\mathcal{C}^{\infty}$-ring homomorphism between von Neumann regular $\mathcal{C}^{\infty}$-rings.
In the present we develop a fragment of the theory of superfields, polynomials and Marshall's quotient in order to obtain for general special groups, a proof of the Arason-Pfister Hauptsatz (APH): "if $ϕ\neq \emptyset$ is an anisotropic form and $ϕ\in I^n(F)$ then $dim (ϕ) \geq 2^n$". In the process, we also obtain an alternative proof of APH for reduced special groups that avoid the uses of the invariants developed in \cite{dickmann2000special}. The applications of the full Arason-Pfister Hauptsatz leads to interesting properties of graded rings associated to special groups/hyperfields. \textbf{Keywords:} Arason-Pfister Hauptsatz; hyperfields; special groups; Milnor K-theory; graded rings.
In this ongoing work, we extend to a class of well-behaved pre-special hyperfields the work of J. Miná\v c and Spira (\cite{minac1996witt}) that describes a (pro-2)-group of a field extension that encodes the quadratic form theory of a given field $F$: in \cite{adem1999cohomology} it is shown that its associated cohomology ring contains a copy of the cohomology ring of the field $F$. Our construction, a contravariant functor into the category of "pointed" pro-2-groups, is essentially given by generators and relations of profinite-2-groups. We prove that such profinite groups $\mbox{Gal}(F)$ encode the space of orders of the special group canonically associated to the hyperfield $F$ and provide a criterion to detect when $F$ is formally real or not.
In this paper, we introduce a new definition of sheaves on semicartesian quantales, providing first examples and categorical properties. We note that our sheaves are similar to the standard definition of a sheaf on a locale, however, we prove in that in general it is not an elementary topos - since the lattice of external truth values of $Sh(Q)$, $Sub(1)$, is canonically isomorphic to the quantale $Q$ - placing this paper as part of a greater project towards a monoidal (not necessarily cartesian) closed version of elementary topos. To start the study the logical aspects of the category of sheaves we are introducing, we explore the nature of the "internal truth value objects" in such sheaves categories. More precisely, we analyze two candidates for subobject classifier for different subclasses of commutative and semicartesian quantales.
The goal of this work is twofold: (i) to provide a detailed analysis of some categories of inductive graded ring - a concept introduced in [DM98] in order to provide a solution of Marshall's signature conjecture in the algebraic theory of quadratic forms; (ii) apply this analysis to deepen the connections between the category of special hyperfields ([dARM22]) - equivalent to the category of special groups ([DM00]) and the categories of inductive graded rings.
Building over some ideas of René Guitart, we provide a categorical framework towards some deviation notions in abstract logic.
Motivated by some recent developments in abstract theories of quadratic forms, we start to develop in this work an expansion of Linear Algebra to multivalued structures (a multialgebraic structure is essentially an algebraic structure but endowed with some multivalued operations). We introduce and study matrices and determinants over a commutative superrings (roughly, a ring where the sum and product are multivalued) and study linear systems and vector spaces over superfields. As an application, we obtain a fundamental result to the development of a theory of algebraic extensions of superfields.
This work mainly concerns the -- here introduced -- category of $\mathscr Q$-sets and functional morphisms, where $\mathscr Q$ is a commutative semicartesian quantale. We describe, in detail, the limits and colimits of this complete and cocomplete category and prove that it has a classifier for regular subobjects. Moreover, we prove that it is $κ^+$-locally presentable category, where $κ=max\{|\mathscr Q|, \aleph_0)\}$ and describe a hierarchy of semicartesian monoidal closed structures in this category. Finally, we discuss the issue of 'change of basis' induced by appropriate morphisms between the parametrizing quantales involved in the definition of $\mathscr Q$-sets. In a future work we will address such questions in the full subcategory given by all Scott-complete $\mathscr Q$-sets
This work is largely focused on extending D. Higgs' $Ω$-sets to the context of quantales, following the broad program of U. Höhle, we explore the rich category of $\mathscr Q$-sets for strong, integral and commutative quantales, or other similar axioms. The focus of this work is to study the different notion of 'completeness' a $\mathscr Q$-set may enjoy and their relations, completion functors, resulting reflective subcategories, their relations to relational morphisms. We establish the general equivalence of singleton complete $\mathscr Q$-sets with functional morphisms and the category of $\mathscr Q$-sets with relational morphisms; we provide two characterizations of singleton completeness in categorical terms; we show that the singleton complete categorical inclusion creates limits.
Building over recent results, we expand the basic theory of algebraic extensions to the realm of superfields -a field with multivalued sum and product-, showing that every superfield has a (unique up to isomorphism) strong algebraic extension to a superfield that is algebraically closed. Moreover we show that every infinite algebraically closed superfield admits quantifier elimination procedure.
We adjust the notion of finitary filter pair, which was coined for creating and analyzing finitary logics, in such a way that we can treat logics of cardinality $κ$, where $κ$ is a regular cardinal. The corresponding new notion is called $κ$-filter pair. A filter pair can be seen as a presentation of a logic, and we ask what different $κ$-filter pairs give rise to a fixed logic of cardinality $κ$. To make the question well-defined we restrict to a subcollection of filter pairs and establish a bijection from that collection to the set of natural extensions of that logic by a set of variables of cardinality $κ$. Along the way we use $κ$-filter pairs to construct natural extensions for a given logic, work out the relationships between this construction and several others proposed in the literature, and show that the collection of natural extensions forms a complete lattice. In an optional section we introduce and motivate the concept of a general filter pair.
The concept of multialgebraic structure -- an "algebraic like" structure but endowed with multiple valued operations -- has been studied since the 1930's; in particular, the concept of hyperrings was introduced by Krasner in the 1950's. Some general algebraic study has been made on multialgebras: see for instance \cite{golzio2018brief} and \cite{pelea2006multialgebras}. More recently the notion of multiring have obtained more attention: a multiring is a lax hyperring, satisfying an weak distributive law, but hyperfields and multifields coincide. Multirings has been studied for applications in abstract quadratic forms theory (\cite{marshall2006real}, \cite{worytkiewiczwitt2020witt}) and tropical geometry (\cite{jun2015algebraic}); a more detailed account of variants of concept of polynomials over hyperrings is even more recent (\cite{jun2015algebraic}, \cite{ameri2019superring}). In the present work we start a model-theoretic oriented analysis of multialgebras introducing the class of algebraically closed and providing variant proof of quantifier elimination flavor, based on new results on superring of polynomials (\cite{ameri2019superring}).
To the best of our knowledge, there are very few results on how Heyting-valued models are affected by the morphisms on the complete Heyting algebras that determine them: the only cases found in the literature are concerning automorphisms of complete Boolean algebras and complete embedding between them (\emph{i.e}., injective Boolean algebra homomorphisms that preserves arbitrary suprema and arbitrary infima). In the present work, we consider and explore how more general kinds of morphisms between complete Heyting algebras $\mathbb{H}$ and $\mathbb{H}'$ induce arrows between $V^{(\mathbb{H})}$ and $V^{(\mathbb{H}')}$, and between their corresponding localic toposes $\mathbf{Set}^{(\mathbb{H})}$ ($\simeq \mathbf{Sh}(\mathbb{H})$) and $\mathbf{Set}^{(\mathbb{H}')}$ ($\simeq \mathbf{Sh}(\mathbb{H}')$). In more details: any {\em geometric morphism} $f^* : \mathbf{Set}^{(\mathbb{H})} \to \mathbf{Set}^{(\mathbb{H'})}$, (that automatically came from a unique locale morphism $f : \mathbb{H} \to \mathbb{H}'$), can be "lifted" to an arrow $\tilde{f} : V^{(\mathbb{H})} \to V^{(\mathbb{H}')}$. We also provide also some semantic preservation results concerning this arrow $\tilde{f} : V^{(\mathbb{H})} \to V^{(\mathbb{H}')}$.
In this paper we state and prove ad hoc "Separation Theorems" of the so-called Smooth Commutative Algebra, the Commutative Algebra of \(\mathcal{C}^{\infty}-\)rings. These results are formally similar to the ones we find in (ordinary) Commutative Algebra. However, their proof is not so straightforward, since it depends on the introduction of the concept of "smooth saturation". As an application of these theorems we present an interesting result that sheds light on the connections between the smooth Zariski spectrum and the real smooth spectrum of a \(\mathcal{C}^{\infty}-\)ring.