Two isomorphism criteria for directed colimits

Using the general notions of finitely presentable and finitely generated object introduced by Gabriel and Ulmer in 1971, we prove that, in any (locally small) category, two sequences of finitely presentable objects and morphisms (or two sequences of finitely generated objects and monomorphisms) have isomorphic colimits (=direct limits) if, and only if, they are confluent. The latter means that the two given sequences can be connected by a back-and-forth chain of morphisms that is cofinal on each side, and commutes with the sequences at each finite stage. In several concrete situations, analogous isomorphism criteria are typically obtained by ad hoc arguments. The abstract results given here can play the useful r\^ole of discerning the general from the specific in situations of actual interest. We illustrate by applying them to varieties of algebras, on the one hand, and to dimension groups---the ordered $K_0$ of approximately finite-dimensional C*-algebras---on the other. The first application encompasses such classical examples as Kurosh's isomorphism criterion for countable torsion-free Abelian groups of finite rank. The second application yields the Bratteli-Elliott Isomorphism Criterion for dimension groups. Finally, we discuss Bratteli's original isomorphism criterion for approximately finite-dimensional C*-algebras, and show that his result does not follow from ours.Comment: 10 page

Representation of Perfect and Local MV-algebras

We describe representation theorems for local and perfect MV-algebras in terms of ultraproducts involving the unit interval [0,1]. Furthermore, we give a representation of local Abelian lattice-ordered groups with strong unit as quasi-constant functions on an ultraproduct of the reals. All the above theorems are proved to have a uniform version, depending only on the cardinality of the algebra to be embedded, as well as a definable construction in ZFC. The paper contains both known and new results and provides a complete overview of representation theorems for such classes

Canonical formulas for k-potent commutative, integral, residuated lattices

Canonical formulas are a powerful tool for studying intuitionistic and modal logics. Actually, they provide a uniform and semantic way to axiomatise all extensions of intuitionistic logic and all modal logics above K4. Although the method originally hinged on the relational semantics of those logics, recently it has been completely recast in algebraic terms. In this new perspective canonical formulas are built from a finite subdirectly irreducible algebra by describing completely the behaviour of some operations and only partially the behaviour of some others. In this paper we export the machinery of canonical formulas to substructural logics by introducing canonical formulas for $k$-potent, commutative, integral, residuated lattices ($k$-$\mathsf{CIRL}$). We show that any subvariety of $k$-$\mathsf{CIRL}$ is axiomatised by canonical formulas. The paper ends with some applications and examples.Comment: Some typo corrected and additional comments adde

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

Stone-Gelfand duality for metrically complete lattice-ordered groups

We extend Yosida's 1941 version of Stone-Gelfand duality to metrically complete unital lattice-ordered groups that are no longer required to be real vector spaces. This calls for a generalised notion of compact Hausdorff space whose points carry an arithmetic character to be preserved by continuous maps. The arithmetic character of a point is (the complete isomorphism invariant of) a metrically complete additive subgroup of the real numbers containing $1$, namely, either $\frac{1}{n}\mathbb{Z}$ for an integer $n = 1, 2, \dots$, or the whole of $\mathbb{R}$. The main result needed to establish the extended duality theorem is a substantial generalisation of Urysohn's Lemma to such "arithmetic" compact Hausdorff spaces. The original duality is obtained by considering the full subcategory of spaces whose each point is assigned the entire group of real numbers. In the introduction we indicate motivations from and connections with the theory of dimension groups.Comment: 24 pages, 2 figure

Social anxiety and Internet gaming disorder: The role of motives and metacognitions

AbstractBackground and aimsIn recent years, Internet Gaming Disorder (IGD) has been recognized as a mental health problem. Although research has found that social anxiety, motives, the preference for online social interactions (POSI), and metacognitions about online gaming are independent predictors of IGD, less is known about their relative contribution to IGD. The aim of the current study was to model the relationship between social anxiety, motives, POSI, metacognitions about online gaming, and IGD.MethodsFive hundred and forty three Italian gamers who play more than 7 h a week (mean age = 23.9 years; SD = 6.15 years; 82.5% males) were included in the study. The pattern of relationships specified by the theoretical model was examined through path analysis.ResultsResults showed that social anxiety was directly associated with four motives (escape, coping, fantasy, and recreation), POSI, and positive and negative metacognitions about online gaming, and IGD. The Sobel test showed that negative metacognitions about online gaming played the strongest mediating role in the relationship between social anxiety and IGD followed by escape, POSI, and positive metacognitions. The model accounted for 54% of the variance for IGD.Discussion and conclusionsOverall, our findings show that, along with motives and POSI, metacognitions about online gaming may play an important role in the association between social anxiety and IGD. The clinical and preventive implications of these findings are discussed

A systematic and critical review of life cycle approaches to assess circular economy pathways in the agri-food sector

This study provides a literature review of life cycle approaches used to assess circular economy (CE) pathways in the agri-food sector. The scope of this review is to understand how and how much the LC-based analysis is useful to evaluate if CE strategies are more sustainable than linear/traditional economic models in agri-food production systems. To carry out the systematic and critical literature review the PRISMA (Preferred Reporting Items for Systematic Reviews and Meta-Analyses) protocol was employed. The literature search was performed employing scientific databases (Scopus and Web of Science). The results highlight that 52 case studies out of 84 (62% of the total) use stand-alone life cycle assessment (LCA) to evaluate the benefits/impacts of circular economy strategies. Only eight studies (9.5%) deal with the life cycle costing (LCC) approach combined with other analyses, while no paper deals with the social life cycle assessment (S-LCA) methodology. We argue that experts in life cycle methodologies must strive to adopt some key elements to ensure that the results obtained fit perfectly with the measurements of circularity and that these can even be largely based on a common basis