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

Poset products as relational models

We introduce a relational semantics based on poset products, and provide sufficient conditions guaranteeing its soundness and completeness for various substructural logics. We also demonstrate that our relational semantics unifies and generalizes two semantics already appearing in the literature: Aguzzoli, Bianchi, and Marra's temporal flow semantics for H\'ajek's basic logic, and Lewis-Smith, Oliva, and Robinson's semantics for intuitionistic Lukasiewicz logic. As a consequence of our general theory, we recover the soundness and completeness results of these prior studies in a uniform fashion, and extend them to infinitely-many other substructural logics

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