Nonstandard fuzzy sets

Abstract Given a set X , we take into consideration the lattice F(X, ∗ [0, 1]) of the nonstandard fuzzy subsets of X , that is the L -subsets with L equal to the unitary interval ∗ [0, 1] of a nonstandard model of analysis. To show the appropriateness of such a concept, we give two examples of vague concepts, positive divergence for functions and vagueness for fuzzy sets, that are representable by suitable nonstandard fuzzy sets. One proves that they are not representable by Zadeh's fuzzy sets. Also, we observe that the same operations and relations defined for fuzzy sets are definable for nonstandard fuzzy sets. In particular, the complementation operation and the sharpening relation. Finally one proves that every L -subset with L totally ordered is a nonstandard fuzzy set.