Composable measures of fuzziness
Abstract
We consider measures of uncertainty in situations of intrinsic ambiguity. We assume that these measures are governed by two composition laws - one dealing with disjoint unions of fuzzy sets, the other with direct products. The composition laws are expressed by means of positive real-valued functions F and G defined on 4-tuples of positive real numbers. Our assumptions lead to a system of five functional equations for F and G. Specifically, the binary operation ⊕ defined by (x, u) ⊕ (y, v) = (x + y, F(x, y, u, v)), for x, y, u, v ∈ (0, + ∞), must be associative and commutative. The same is true of the binary operation ○ defined by (x, u) ○ (y, v) = (xy, G(x, y, u, v)). Finally, the operation ○ must distribute over ⊕. We determine all pairs of functions F, G which are continuous and strictly monotonic in their last two variables and which satisfy the system of functional equations described above.
