Tense distributive lattices: algebra, logic and topology

Tense logic was introduced by Arthur Prior in the late 1950s as a result of his interest in the relationship between tense and modality. Prior's idea was to add four primitive modal-like unary connectives to the base language today widely known as Prior's tense operators. Since then, Prior's operators have been considered in many contexts by different authors, in particular, in the context of algebraic logic. Here, we consider the category tdlat of bounded distributive lattices equipped with Prior's tense operators. We establish categorical dualities for tdlat in terms of certain categories of Kripke frames and Priestley spaces, respectively. As an application, we characterize the congruence lattice of any tense distributive lattice as well as the subdirectly irreducible members of this category. Finally, we define the logic that preserves degrees of truth with respect to tdlat-algebras and precise the relation between particular sub-classes of tdlat and know tense logics found in the literature

Subdirectly Irreducible IKt-Algebras

The IKt-algebras that we investigate in this paper were introduced in the paper "An algebraic axiomatization of the Ewald’s intuitionistic tense logic" by the first and third author. Now we characterize by topological methods the subdirectly irreducible IKt-algebras and particularly the simple IKt-algebras. Finally, we consider the particular cases of finite IKt-algebras and complete IKt-algebras.Fil: Figallo, Aldo Victorio. Universidad Nacional de San Juan. Facultad de Filosofía, Humanidades y Artes. Instituto de Ciencias Básicas; ArgentinaFil: Pascual, Inés Beatriz. Universidad Nacional de San Juan. Facultad de Filosofía, Humanidades y Artes. Departamento de Matemática; Argentina. Universidad Nacional de San Juan. Facultad de Filosofía, Humanidades y Artes. Instituto de Ciencias Básicas; ArgentinaFil: Pelaitay, Gustavo Andrés. Universidad Nacional de San Juan. Facultad de Filosofía, Humanidades y Artes. Departamento de Matemática; Argentina. Universidad Nacional de San Juan. Facultad de Filosofía, Humanidades y Artes. Instituto de Ciencias Básicas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - San Juan; Argentin

An Algebraic Study of Tense Operators on Nelson Algebras

Ewald considered tense operators G, H, F and P on intuitionistic propositional calculus and constructed an intuitionistic tense logic system called IKt. In 2014, Figallo and Pelaitay introduced the variety IKt of IKt-algebras and proved that the IKt system has IKt-algebras as algebraic counterpart. In this paper, we introduce and study the variety of tense Nelson algebras. First, we give some examples and we prove some properties. Next, we associate an IKt-algebra to each tense Nelson algebras. This result allowed us to determine the congruence of the tense Nelson algebras and also to characterize the subdirectly irreducible tense Nelson algebras and particularly the simple tense Nelson algebras. Finally, we prove that there exists an equivalence between the category of IKt-algebras and the category of tense centered Nelson algebras.Fil: Figallo, Aldo Victorio. Universidad Nacional de San Juan. Facultad de Filosofía, Humanidades y Artes. Instituto de Ciencias Básicas; ArgentinaFil: Pelaitay, Gustavo Andrés. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de San Juan. Facultad de Filosofía, Humanidades y Artes. Instituto de Ciencias Básicas; ArgentinaFil: Sarmiento, Jonathan Matias. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca; Argentina. Universidad Nacional del Sur. Departamento de Matemática; Argentin