On the closure of families of fuzzy measures under eventwise aggregations

Necessary and sufficient consensus conditions for the eventwise aggregation of lower probabilities

“…First we recall some results obtained in [2], then we investigate families of aggregation functions by the technique, which is similar to well-known constructions like the condition "avoiding sure loss" or "natural extension" in the theory of imprecise probabilities [11], and finally we prove that the sufficient conditions, found in [2] for aggregation functions, are also necessary ones.…”

“…A function : [0, 1] n → [0, 1] is called n-aggregation function (or n-aggregation operator [2,3]) if • g ∈ M 0 for any g = (g 1 , . .…”

“…Some recent results are given in [2], where (generalized) coherent lower probabilities and k-monotone measures are of interest. The pointed paper gives necessary and sufficient conditions for aggregation functions if 1 , .…”

On extension of fuzzy measures to aggregation functions

“…For example, by Theorem 2, for a binary aggregation function A with zero annihilator the function F m,A is non-decreasing for each m ∈ M (2) if and only if A-volumes of all possible 2-boxes…”

“…, which in binary case are also obtained from conditions (2), are trivially nonnegative because of the monotonicity of A.…”