Scalar and fuzzy cardinalities of crisp and fuzzy multisets

Abstract

We define in an axiomatic way scalar and fuzzy cardinalities of finite multisets over ]0, 1], and we obtain explicit descriptions for them. We show that, for multisets over ]0, 1] associated to finite fuzzy sets, the cardinalities defined in this study are equivalent to the cardinalities of the corresponding fuzzy sets previously introduced in the literature. Finally, we also define in an axiomatic way scalar and fuzzy cardinalities of finite fuzzy multisets over any set X, and we use the descriptions of the cardinalities of finite multisets over ]0, 1] to obtain explicit characterizations of the former. © 2009 Wiley Periodicals, Inc.