SMETS-MAGREZ AXIOMS FOR R-IMPLICATORS IN INTERVAL-VALUED AND INTUITIONISTIC FUZZY SET THEORY
Abstract
Interval-valued fuzzy sets constitute an extension of fuzzy sets which give an interval approximating the "real" (but unknown) membership degree. Interval-valued fuzzy sets are equivalent to intuitionistic fuzzy sets in the sense of Atanassov which give both a membership degree and a non-membership degree, whose sum must be smaller than or equal to 1. Both are equivalent to L-fuzzy sets w.r.t. a special lattice L*. In fuzzy set theory, an important class of triangular norms is the class of those that satisfy the residuation principle. In a previous paper5 we gave a construction for t-norms on L* satisfying the residuation principle which are not t-representable. In this paper we investigate the Smets-Magrez axioms and some other properties for the residual implicator generated by such t-norms.
