Uniform-type structures on lattice-valued spaces and frames

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...

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Bibliographic Details
Published in:Fuzzy sets and systems 2008-10, Vol.159 (19), p.2469-2487
Main Authors: Gutiérrez García, Javier, Mardones-Pérez, Iraide, Picado, Jorge, de Prada Vicente, María Angeles
Format: Article
Language:English
Subjects:
Axiality
Cover
Entourage
Frame
Galois connection
Girard quantale
Integral commutative quantale
L-valued frame
L-valued space
Locale
Polarity
Quantale
Uniform operator
Uniformity
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Summary: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 of L-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).
ISSN:0165-0114
1872-6801
DOI:10.1016/j.fss.2008.03.004