A note on drastic product logic

The drastic product $*_D$ is known to be the smallest $t$-norm, since $x *_D y = 0$ whenever $x, y < 1$. This $t$-norm is not left-continuous, and hence it does not admit a residuum. So, there are no drastic product $t$-norm based many-valued logics, in the sense of [EG01]. However, if we renounce standard completeness, we can study the logic whose semantics is provided by those MTL chains whose monoidal operation is the drastic product. This logic is called ${\rm S}_{3}{\rm MTL}$ in [NOG06]. In this note we justify the study of this logic, which we rechristen DP (for drastic product), by means of some interesting properties relating DP and its algebraic semantics to a weakened law of excluded middle, to the $\Delta$ projection operator and to discriminator varieties. We shall show that the category of finite DP-algebras is dually equivalent to a category whose objects are multisets of finite chains. This duality allows us to classify all axiomatic extensions of DP, and to compute the free finitely generated DP-algebras.Comment: 11 pages, 3 figure

Stone-type representations and dualities for varieties of bisemilattices

In this article we will focus our attention on the variety of distributive bisemilattices and some linguistic expansions thereof: bounded, De Morgan, and involutive bisemilattices. After extending Balbes' representation theorem to bounded, De Morgan, and involutive bisemilattices, we make use of Hartonas-Dunn duality and introduce the categories of 2spaces and 2spaces$^{\star}$. The categories of 2spaces and 2spaces$^{\star}$ will play with respect to the categories of distributive bisemilattices and De Morgan bisemilattices, respectively, a role analogous to the category of Stone spaces with respect to the category of Boolean algebras. Actually, the aim of this work is to show that these categories are, in fact, dually equivalent

Expressivity and correspondence theory of many-valued hybrid logic

Abstract: The aim of this dissertation is to identify the construction of models that preserve (in both directions) the truth of hybrid formulas and therefore serve to characterize the expressivity of many-valued hybrid logic based on the framework of Hansen, Bolander and Brauner. We show that generated submodels and bounded morphisms preserve the truth of hybrid formulas in both directions. We also show that bisimilarity implies hybrid equivalence in general, however, the converse is not true in general. The converse is true for a weaker notion of a bisimulation for a special set of models, the image-finite models. The second significant contribution of this project is to develop the correspondence theory for many-valued hybrid logic. We show that the algorithm ALBA(first developed by Conradie and Palmigiano) can be extended to the many-valued hybrid setting. We call this extension MV-Hybrid ALBA. As a result, we successfully identify a syntactically defined class of hybrid formulas for a many-valued hybrid language, namely inductive formulas, whose members always have a local first-order frame correspondents. This inductive class generalizes the Sahlqvist class. An appropriate duality is obtained between frames in the chosen many-valued hybrid framework and a class of algebras having certain properties in order to extend ALBA to the many-valued hybrid setting.M.Sc. (Applied Mathematics