10 research outputs found
Orders on groups, and spectral spaces of lattice-groups
No full textExtending pioneering work by Weinberg, Conrad, McCleary, and others, we provide a systematic way of relating spaces of right orders on a partially ordered group, on the one hand, and spectral spaces of free lattice-ordered groups, on the other. The aim of the theory is to pave the way for further fruitful interactions between the study of right orders on groups and that of lattice-groups. Special attention is paid to the important case of orders on groups
Ordering Free Groups and Validity in Lattice-Ordered Groups
No full textAn inductive characterization is given of the subsets of a group that extend to the positive cone of a right order on the group. This characterization is used to relate validity of equations in lattice-ordered groups (l-groups) to subsets of free groups that extend to the positive cone of a right order. As a consequence, new proofs are obtained of the decidability of the word problem for free l-groups and generation of the variety of l-groups by the l-group of automorphisms of the real line. An inductive characterization is also given of the subsets of a group that extend to the positive cone of an order on the group. In this case, the characterization is used to relate validity of equations in varieties of representable l-groups to subsets of relatively free groups that extend to the positive cone of an order
Proof theory for positive logic with weak negation
No full textProof-theoreticmethods are developed for subsystems of Johansson’s logic obtained by extending the positive fragment of intuitionistic logic with weak negations. These methods are exploited to establish properties of the logical systems. In particular, cut-free complete sequent calculi are introduced and used to provide a proof of the fact that the systems satisfy the Craig interpolation property. Alternative versions of the calculi are later obtained by means of an appropriate loop-checking history mechanism. Termination of the new calculi is proved, and used to conclude that the considered logical systems are PSPACE-complete
Proof Theory and Ordered Groups
No full textOrdering theorems, characterizing when partial orders of a group extend to total orders, are used to generate hypersequent calculi for varieties of lattice-ordered groups (ℓ-groups). These calculi are then used to provide new proofs of theorems arising in the theory of ordered groups. More precisely: an analytic calculus for abelian ℓ-groups is generated using an ordering theorem for abelian groups; a calculus is generated for ℓ-groups and new decidability proofs are obtained for the equational theory of this variety and extending finite subsets of free groups to right orders; and a calculus for representable ℓ-groups is generated and a new proof is obtained that free groups are orderable
Theorems of Alternatives for Substructural Logics
No full textA theorem of alternatives provides a reduction of validity in a substructural logic to validity in its multiplicative fragment. Notable examples include a theorem of Arnon Avron that reduces the validity of a disjunction of multiplicative formulas in the "R-mingle" logic RM to the validity of a linear combination of these formulas, and Gordan’s theorem for solutions of linear systems over the real numbers that yields an analogous reduction for validity in Abelian logic A. In this paper, general conditions are provided for axiomatic extensions of involutive uninorm logic without additive constants to admit a theorem of alternatives. It is also shown that a theorem of alternatives for a logic can be used to establish (uniform) deductive interpolation and completeness with respect to a class of dense totally ordered residuated lattices
From L-Groups to Distributive L-Monoids, and Back Again
No full textWe prove that an inverse-free equation is valid in the variety LG of lattice-ordered groups (L-groups) if and only if it is valid in the variety DLM of distributive latticeordered monoids (distributive L-monoids). This contrasts with the fact that, as proved by Repnitskii, there exist inverse-free equations that are valid in all Abelian L-groups but not in all commutative distributive L-monoids, and, as we prove here, there exist inverse-free equations that are valid in all totally ordered groups but not in all totally ordered monoids. We also prove that DLM has the finite model property and a decidable equational theory, establish a correspondence between the validity of equations in DLM and the existence of certain right orders on free monoids, and provide an effective method for reducing the validity of equations in LG to the validity of equations in DLM
Subminimal negation
No full textMinimal logic, i.e., intuitionistic logic without the ex falso principle, is investigated in its original form with a negation symbol instead of a symbol denoting the contradiction. A Kripke semantics is developed for minimal logic and its sublogics with a still weaker negation by introducing a function on the upward closed sets of the models. The basic logic is a logic in which the negation has no properties but the one of being a unary operator. A number of extensions is studied of which the most important ones are contraposition logic and negative ex falso, a weak form of the ex falso principle. Completeness is proved, and the created semantics is further studied. The negative translation of classical logic into intuitionistic logic is made part of a chain of translations by introducing translations from minimal logic into contraposition logic and intuitionistic logic into minimal logic, the latter having been discovered in the correspondence between Johansson and Heyting. Finally, as a bridge to the work of Franco Montagna a start is made of a study of linear models of these logics
Order, algebra, and structure: lattice-ordered groups and beyond
No full textThis thesis describes and examines some remarkable relationships existing between seemingly quite different properties (algebraic, order-theoretic, and structural) of ordered groups. On the one hand, it revisits the foundational aspects of the structure theory of lattice-ordered groups, contributing a novel systematization of its relationship with the theory of orderable groups. One of the main contributions in this direction is a connection between validity in varieties of lattice-ordered groups, and orders on groups; a framework is also provided that allows for a systematic account of the relationship between orders and preorders on groups, and the structure theory of lattice-ordered groups. On the other hand, it branches off in new directions, probing the frontiers of several different areas of current research. More specifically, one of the main goals of this thesis is to suitably extend results that are proper to the theory of lattice-ordered groups to the realm of more general, related algebraic structures; namely, distributive lattice-ordered monoids and residuated lattices. The theory of lattice-ordered groups provides themain source of inspiration for this thesis’ contributions on these topics
A Study of Subminimal Logics of Negation and Their Modal Companions
No full textWe study propositional logical systems arising from the language of Johansson's minimal logic and obtained by weakening the requirements for the negation operator. We present their semantics as a variant of neighbourhood semantics. We use duality and completeness results to show that there are uncountably many subminimal logics. We also give model-theoretic and algebraic definitions of filtration for minimal logic and show that they are dual to each other. These constructions ensure that the propositional minimal logic has the finite model property. Finally, we define and investigate bi-modal companions with non-normal modal operators for some relevant subminimal systems, and give infinite axiomatizations for these bi-modal companions
Dynamic comparison of portfolio risk: Clean vs dirty energy
No full textThis paper analyses whether investing in clean energy significantly worsens the risk level of investors. To that aim, we propose a dynamic strategy to carry out a comparative risk analysis of three minimum-variance portfolios: a portfolio made up exclusively of dirty energies, a portfolio made up only of clean energy assets, and a portfolio combined with the two types of energies. To that aim, we use multivariate GARCH models, concretely Asymmetric Dynamic Conditional Correlations models (ADCC-GARCH) to predict the variance and covariance matrices of the daily asset returns and we compare the portfolio volatilities using the methodology proposed by Engle and Colacito (2006). The analysed period was from January 2010 to September 2021, so that the data include half of phase II, full phase III and the onset of phase IV of the EU ETS, as well as the Brexit and COVID-19 outbreaks in the European context. Our results show that, unlike what happened in other economic crises (subprime, Brexit), from the pandemic crisis, the investment in clean energies is preferable to fossil energies, not only in terms of profitability, as other studies have shown, but also in terms of risk. Therefore, investing in clean energy companies, which are aligned with their role towards socially responsible initiatives, is valuable not only for its contribution to a sustainable energy transition to renewable sources but also for the attractiveness from a financial point of view. © 2022 The Author(s
