W-algebras which are Boolean products of members of SR[1] and CW-algebras

We show that the class of all isomorphic images of Boolean Products of members of SR [1] is the class of all archimedean W-algebras. We obtain this result from the characterization of W-algebras which are isomorphic images of Boolean Products of CW-algebras.     

Axiomatic Extensions of IMT3 Logic

In this paper we characterize, classify and axiomatize all axiomatic extensions of the IMT3 logic. This logic is the axiomatic extension of the involutive monoidal t-norm logic given by ¬φ³ ∨ φ. For our purpose we study the lattice of all subvarieties of the class IMT3, which is the variety of IMTL-algebras given by the equation ¬(x³) ∨ x ≈ ?, and it is the algebraic counterpart of IMT3 logic. Since every subvariety of IMT3 is generated by their totally ordered members, we study the structure of all IMT3-chains in order to determine the lattice of all subvarieties of IMT3. Given a family of IMT3-chains the number of elements of the largest odd finite subalgebra in the family and the number of elements of the largest even finite subalgebra in the family turns out to be a complete classifier of the variety generated. We obtain a canonical set of generators and a finite equational axiomatization for each subvariety and, for each corresponding logic, a finite set of characteristic matrices and a finite set of axioms.     

Quasivarieties Generated by Simple MV-algebras

In this paper we show that the quasivariety generated by an infinite simple MV-algebra only depends on the rationals which it contains. We extend this property to arbitrary families of simple MV-algebras.     

Measuring the core features of personality disorders in substance abusers using the Temperament and Character Inventory (TCI)

Personality disorders (PDs) are still classified through categorical taxonomies that are at odds with current research findings. Dimensional models provide a suitable alternative for measuring individual differences. However, as they have traditionally lacked a clear definition of the "disorder" construct, the clinical utility of these models has been limited. This study tests whether Cloninger's dimensional model is able to capture two domains: the features that differentiate PD subtypes from each other and the common core features underlying all PDs. Seventy-four drug dependent patients were independently assessed using the SCID-II and Cloninger's TCI. There was a slight relationship between TCI temperament dimensions and the DSM personality disorder subtypes, but the association was not specific enough to allow differential diagnosis. The character dimension Self-Directedness was strongly associated with the presence and severity of all PDs, irrespective of subtype, correctly classifying 77% of subjects. Character dimensions are a reliable, valid and low-cost tool for detecting PDs in drug abusers and may help to provide an operational definition of the common core features underlying all PDs.     

Semisimples in Varieties of Commutative Integral Bounded Residuated Lattices

In any variety of bounded integral residuated lattice-ordered commutative monoids (bounded residuated lattices for short) the class of its semisimple members is closed under isomorphic images, subalgebras and products, but it is not closed under homomorphic images, and so it is not a variety. In this paper we study varieties of bounded residuated lattices whose semisimple members form a variety, and we give an equational presentation for them. We also study locally representable varieties whose semisimple members form a variety. Finally, we analyze the relationship with the property “to have radical term”, especially for k-radical varieties, and for the hierarchy of varieties (WL_k)_k>0 defined in Cignoli and Torrens (Studia Logica 100:1107–1136, 2012 [7]).     

An Approach to Glivenko’s Theorem in Algebraizable Logics

In a classical paper [15] V. Glivenko showed that a proposition is classically demonstrable if and only if its double negation is intuitionistically demonstrable. This result has an algebraic formulation: the double negation is a homomorphism from each Heyting algebra onto the Boolean algebra of its regular elements. Versions of both the logical and algebraic formulations of Glivenko’s theorem, adapted to other systems of logics and to algebras not necessarily related to logic can be found in the literature (see [2, 9, 8, 14] and [13, 7, 14]). The aim of this paper is to offer a general frame for studying both logical and algebraic generalizations of Glivenko’s theorem. We give abstract formulations for quasivarieties of algebras and for equivalential and algebraizable deductive systems and both formulations are compared when the quasivariety and the deductive system are related. We also analyse Glivenko’s theorem for compatible expansions of both cases. Presented by Jacek Malinowski