Fuzzy Sets Over the Poset I

Abstract

The author studies fuzzy sets over the poset I = [0, 1] with the usual
order. These form a canonical example of fuzzy sets over a poset discussed in (Tiryaki, ˙I. U. and Brown, L. M. Plain textures and fuzzy
sets via posets, preprint). Characterizations of these so called “soft
fuzzy sets” are obtained, and soft fuzzy sets are shown to have a richer
mathematical theory than classical I-fuzzy sets. In particular soft fuzzy
points behave like the points of crisp set theory with respect to join, and
moreover there exists a Lowen type functor from Top to the construct
SF-Top that preserves both separation and compactness.

Keywords

• Texture,Unit interval texture,Hutton algebra,Fuzzy subset,Soft fuzzy subset,Point,Copoint,Construct,SF-topology,Ditopology,Separation,Compactness,Generalized Lowen functor,Rotund soft fuzzy set,Lowen rotund functor,Preservation of topological properties

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