Fuzzy uniform structures and continuous t-norms

Authors:
J. Gutiérrez García;S. Romaguera;M. Sanchis
Affiliations:
Departamento de Matemáticas, Universidad del País Vasco-Euskal Herriko Unibertsitatea, Apartado 644, 48080 Bilbao, Spain;Instituto Universitario de Matemática Pura y Aplicada, Universidad Politécnica de Valencia, 46071 Valencia, Spain;Departament de Matemàtiques, Universitat Jaume I, Campus del Riu Sec. s/n, 12071 Castelló, Spain
Venue:
Fuzzy Sets and Systems
Year:
2010

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Abstract

We introduce and discuss a notion of fuzzy uniform structure that provides a direct link with the classical theory of uniform spaces. More exactly, for each continuous t-norm we prove that the category of all fuzzy uniform spaces in our sense (and fuzzy uniformly continuous mappings) is isomorphic to the category of uniform spaces (and uniformly continuous mappings) by means of a covariant functor. Moreover, we also describe the inverse functor and, then, we discuss completeness and completion of these fuzzy uniform structures. It follows from our results that each fuzzy uniform structure in our sense induces a Hutton [0, 1]-uniformity and, conversely, each Hutton [0, 1]-uniformity induces a fuzzy uniform structure in our sense.