Uniform-type structures on lattice-valued spaces and frames

• Autor(a) principal: Gutiérrez García, Javier
• Data de Publicação: 2008
• Outros Autores: Mardones-Pérez, Iraide, Picado, Jorge, Prada Vicente, María Angeles de
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
• Texto Completo: http://hdl.handle.net/10316/4595; https://doi.org/10.1016/j.fss.2008.03.004
• topic: Quantale; Girard quantale; Integral commutative quantale; L-valued space; L-valued frame; Uniformity; Entourage; Uniform operator; Galois connection; Axiality; Polarity

By introducing lattice-valued covers of a set, we present a general framework for uniform structures on very general L-valued spaces (for L an integral commutative quantale). By showing, via an intermediate L-valued structure of uniformity, how filters of covers may describe the uniform operators of Hutton, we prove that, when restricted to Girard quantales, this general framework captures a significant class of Hutton's uniform spaces. The categories ofL-valued uniform spaces and L-valued uniform frames here introduced provide (in the case L is a complete chain) the missing vertices in the commutative cube formed by the classical categories of topological and uniform spaces and their corresponding pointfree counterparts (forming the base of the cube) and the corresponding L-valued categories (forming the top of the cube).