4 research outputs found
Monotonic Distributive Semilattices
In the study of algebras related to non-classical logics, (distributive) semilattices are always present in the background. For example, the algebraic semantic of the {→, ∧, ⊤}-fragment of intuitionistic logic is the variety of implicative meet-semilattices (Chellas 1980; Hansen 2003). In this paper we introduce and study the class of distributive meet-semilattices endowed with a monotonic modal operator m. We study the representation theory of these algebras using the theory of canonical extensions and we give a topological duality for them. Also, we show how our new duality extends to some particular subclasses.Fil: Celani, Sergio Arturo. Consejo Nacional de Investigaciones CientÃficas y Técnicas; Argentina. Universidad Nacional del Centro de la Provincia de Buenos Aires. Facultad de Ciencias Exactas. Departamento de Matemática; ArgentinaFil: Menchón, MarÃa Paula. Universidad Nacional del Centro de la Provincia de Buenos Aires. Facultad de Ciencias Exactas. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones CientÃficas y Técnicas; Argentin
Polygraphs: From Rewriting to Higher Categories
Polygraphs are a higher-dimensional generalization of the notion of directed
graph. Based on those as unifying concept, this monograph on polygraphs
revisits the theory of rewriting in the context of strict higher categories,
adopting the abstract point of view offered by homotopical algebra. The first
half explores the theory of polygraphs in low dimensions and its applications
to the computation of the coherence of algebraic structures. It is meant to be
progressive, with little requirements on the background of the reader, apart
from basic category theory, and is illustrated with algorithmic computations on
algebraic structures. The second half introduces and studies the general notion
of n-polygraph, dealing with the homotopy theory of those. It constructs the
folk model structure on the category of strict higher categories and exhibits
polygraphs as cofibrant objects. This allows extending to higher dimensional
structures the coherence results developed in the first half