Abstract
An image of a set that produces a multiset from an ordinary set is proposed and contrasted with the ordinary image. Let X={a, b, c, d}, A={a, b, c}, B={a, a, b, b, c}, and C={(a, 0.1), (a, 0.2), (b, 1), (c, 0.5)}.Then A is an ordinary set, B is a crisp multiset, and C is a fuzzy multiset. Assume Y={v, w}and define f:X →Y by f(a)=f(b)=v and f(c)=w. From an ordinary extension of the image f(A)=f(B)=Y and f(C)={(v, 1), (w, 0.5)}, i.e., we do not obtain a multiset. In this paper we introduce another image, denoted by f[A], as follows. For each input x⋴A, f(x)is added to the output regardless whether or not there already exists f(x) in the output. For the above example, f[A]={f(a), f(b), f(c)}={v, v, w}and f[C]={(v, 0.1), (v, 0.2), (v, 1), (w, 0.5)}. Theoretical properties such as commutativity zof f[・]with α-cut and multiset addition are proved. Two applications are studied: one is a rough approximation using a natural projection onto the equivalence classes; the other is a query language in fuzzy relational databases.
