Free fuzzy modules and their bases (Q1310913)

The notions of a basis for fuzzy modules and of free fuzzy modules are defined. The basic properties of these concepts are determined. It is shown that a fuzzy module possesses a basis if and only if it is free. Results on the existence of a basis are also determined. An interesting example is given showing that fuzzy subspaces of a finite dimensional vector space may fail to have a basis. This can happen when no restriction is placed on the lattice of truth values. However, if the lattice of truth values is totally ordered, then every fuzzy subspace of a finite dimensional vector space has a basis.