Some modal and temporal translations of generalized basic logic

We introduce a family of modal expansions of Łukasiewicz logic that are designed to accommodate modal translations of generalized basic logic (as formulated with exchange, weakening, and falsum). We further exhibit algebraic semantics for each logic in this family, in particular showing that all of them are algebraizable in the sense of Blok and Pigozzi. Using this algebraization result and an analysis of congruences in the pertinent varieties, we establish that each of the introduced modal Łukasiewicz logics has a local deduction-detachment theorem. By applying Jipsen and Montagna’s poset product construction, we give two translations of generalized basic logic with exchange, weakening, and falsum in the style of the celebrated Gödel-McKinsey-Tarski translation. The first of these interprets generalized basic logic in a modal Łukasiewicz logic in the spirit of the classical modal logic S4, whereas the second interprets generalized basic logic in a temporal variant of the latter

Split exact sequences of finite MTL-chains

This paper is devoted to presenting ordinal sums of MTL-chains as a particular case of split short exact sequences of finite chains in the category of semihoops. This module theoretical approach will allows us to prove, in a very elementary way, that every finite locally unital MTL-chain can be decomposed as an ordinal sum of archimedean MTL-chains. Furthermore, we propose the study of MTL-chain extensions and we show that ordinal sums of locally unital MTL-chains are a particular case of these.Fil: Castiglioni, José Luis. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata; Argentina. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; ArgentinaFil: Zuluaga Botero, William Javier. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata; Argentina. Universidad Nacional del Centro de la Provincia de Buenos Aires. Facultad de Ciencias Exactas. Departamento de Matemática; Argentin

On finite MTL-algebras that are representable as poset products of archimedean chains

We obtain a duality between the category of locally unital finite MTL-algebras and the category of finite labeled trees. In addition we prove that certain poset products of MTL-algebras are in fact, sheaves of MTL-chains over Alexandrov spaces. Finally we give a concrete description for the studied poset products in terms of direct products and ordinal sums of finite MTL-algebras.Fil: Castiglioni, José Luis. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata; ArgentinaFil: Zuluaga Botero, William Javier. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata; Argentin

A representation theorem for integral rigs and its applications to residuated lattices

We generalize the Dubuc–Poveda representation theorem for MV-algebras so that it applies to other algebraic categories of residuated join-semilattices. In particular, as a corollary, we obtain a representation result for pre-linear residuated join-semilattices in terms of totally ordered fibers. The main result is analogous to the Zariski representation of (commutative) rings and it is proved using tools from topos theory.Facultad de Ciencias Exacta

A representation theorem for integral rigs and its applications to residuated lattices

We generalize the Dubuc–Poveda representation theorem for MV-algebras so that it applies to other algebraic categories of residuated join-semilattices. In particular, as a corollary, we obtain a representation result for pre-linear residuated join-semilattices in terms of totally ordered fibers. The main result is analogous to the Zariski representation of (commutative) rings and it is proved using tools from topos theory.Facultad de Ciencias Exacta

Relatively compatible operations in BCK-algebras and some related algebras

Let A be a BCK-algebra and f : Ak → A a function. The main goal of this paper is to give a necessary and sufficient condition for ƒ to be compatible with respect to every relative congruence of A. We extend this result in some related algebras, as for example in pocrims.Facultad de Ciencias Exacta

Representaciones por haces de riRigs

El presente trabajo está motivado por los teoremas de representación para anillos conmutativos con unidad, obtenidos por Pierce y los desarrollados por Dubuc-Poveda para MV-álgebras; así como las ideas estratégicas de Lawvere sobre el análisis de categorías coextensivas con herramientas de teoría de topos. Los resultados principales de este trabajo son dos. El primero, es una teoría de representación para riRigs como riRigs conexos internos en un topos de haces sobre un álgebra de Boole. El segundo, es una teoría de representación para riRigs como riRigs really local internos en un topos de haces sobre un retículo distributivo. Las construcciones elaboradas para la obtención de las representaciones anteriores no requieren del uso de herramientas topológicas, a diferencia de las representaciones citadas previamente. La traducción de estos resultados en términos de homeos locales sobre espacios espectrales, permite relacionar la representación de Dubuc-Poveda para MV-álgebras con la representación really local de riRigs.Facultad de Ciencias Exacta

Relatively compatible operations in BCK-algebras and some related algebras

Let A be a BCK-algebra and f :A k →A a function. The main goal of this article is to give a necessary and sufficient condition for f to be compatible with respect to every relative congruence of A. We extend this result in some related algebras, as e.g. in pocrims.Fil: Lubomirsky, Noemí. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata; Argentina. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; ArgentinaFil: San Martín, Hernán Javier. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata; ArgentinaFil: Zuluaga Botero, William Javier. Universidad Nacional de Córdoba. Facultad de Ciencias Exactas, Físicas y Naturales. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba; Argentin