Pedrycz shadowed sets are three-way approximations of fuzzy sets by transforming the infinite levels of fuzzy set membership grades in the unit interval [ 0 , 1 ] into three levels. The three levels represent qualitatively the sets of the white, grey, and black members of a shadowed set. In this paper, we generalize the notion of shadowed sets to the case of L-fuzzy sets by making three new contributions. First, we consider two representations of a shadowed set. One is a three-valued L-fuzzy set and the other is three pairwise disjoint sets. Second, we introduce two methods for constructing a shadowed set. One divides a finite lattice based on the notion of a pair of a set of designated core membership grades and a set of designated null membership grades. The other uses a pair of threshold sets, which generalizes the method that uses a pair of thresholds. We study formal properties of the two methods and show that they are equivalent. Finally, based on a distance function on a lattice, we present a simple method to build the sets of designated core and null membership grades.

Recently, Yager introduced several interesting information-theoretic measures of diversity related to the decision problem of selecting multiple objects from a pool of candidates lying in multiple categories. To give decision makers more choice, we introduce two more measures of diversity for the decisions in this note. One is based on the Simpson's diversity index extensively used in ecology, and the other is based on the Jain's fairness index proposed in network engineering community. © 2011 Wiley Periodicals, Inc.