Fuzzy uniform structures and continuous t-norms

doi:10.1016/j.fss.2009.09.021

Fuzzy Sets and Systems

Volume 161, Issue 7, 1 April 2010, Pages 1011-1021

https://doi.org/10.1016/j.fss.2009.09.021 Get rights and content

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.

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^☆: This research is supported by the Plan Nacional $I + D + i$ and FEDER, under Grants MTM2006-14925-C02-01 and MTM2006-14925-C02-02.

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Fuzzy uniform structures and continuous t-norms☆

Abstract

Fuzzy Sets and Systems

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J. Math. Anal. Appl.

J. Math. Anal. Appl.

J. Math. Anal. Appl.

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Fuzzy Sets and Systems