Abstract
This paper considers fuzzy subbundles of a vector bundle. We define the operations sum, product, tensor product, Hom, and intersection of fuzzy subbundles and in each case, we characterize the corresponding flag of vector subbundles. We then propose two alternative definitions of integrability on fuzzy subbundles of a given type and discuss their naturality, merits, and shortcomings. We do these here with a view to introduce and study integrable fuzzy subbundles of tangent bundles on manifolds and foliations in further papers.
Highlights
A fuzzy subspace of a vector space can be characterized as a flag with weights, which are real numbers taken from the unit interval, attached at each component of the “nested” subspaces of the flag [7]
The present work was suggested by analogy with basic constructions of vector bundles
We will have to deal with weights which are new components in the case of fuzzy subbundles
Summary
A fuzzy subspace of a vector space can be characterized as a flag with weights, which are real numbers taken from the unit interval, attached at each component of the “nested” subspaces of the flag [7]. This construction becomes more explicit for a weighted flag of a given type of a vector space of finite dimension over the field F , where F is the field of real or complex numbers.
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