Deep Properties of Totally Fuzzy Sets

It was suggested in [J. Barone (2006)] that totally fuzzy sets could be transformed into "equivalent" ordinary fuzzy sets (totally fuzzy sets where all pairs of elements (x, y) are mapped to zero unless x = y) by choosing an appropriate singleton and then solving a suitable relational equation. This paper describes another method for accomplishing this transformation, namely, by taking the ordinary fuzzy set to be given by the spectrum of eigenvalues of the underlying totally fuzzy set. Special conditions are required to preserve the category-theoretic relationship between the two fuzzy sets, and these are also discussed. Once the process has been outlined, a number of possible areas of application are adumbrated. These include decision theory, linguistic hedges, and the cognitive/linguistic representation of color terms. The conclusion is that totally fuzzy sets and their spectra may have cognitive significance.