Sugeno Utility Functions II: Factorizations

Pseudo-polynomial Functions over Finite Distributive Lattices

“…In this paper we extend the authors' previous works [6,7] by considering an aggregation model f : X 1 × · · · × Xn → Y for arbitrary sets X 1 , . .…”

“…This theorem appears in [7,8] for the special case of chains; for the sake of self-containedness, we reproduce the proof here. We use the notation 1 I for the characteristic vector of I ⊆ [n] in i∈[n] X i , i.e., 1 I ∈ i∈[n] X i is the n-tuple whose i-th component is 1 Xi if i ∈ I, and 0 Xi otherwise.…”

“…For further background on aggregation functions and their use in decision making, we refer the reader to [2,13]; for basics in the theory of lattices, see [10,14]. In Section 3 we develop a general framework used to derive an axiomatization of pseudo-polynomial functions of somewhat different nature than those proposed in [4,6,7], and which will provide tools for determining all possible factorizations of given pseudo-polynomial functions in Section 4. These results are then illustrated in Section 5 by means of a concrete example, and in Section 6 we show how this new procedure can be applied to derive the algorithm provided in [7,8].…”

Pseudo-polynomial functions over finite distributive lattices

“…

In this paper we extend the authors' previous works [6,7] by considering an aggregation model f : X 1 × · · · × Xn → Y for arbitrary sets X 1 , . .

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“…where p is an n-variable lattice polynomial function over Y , and each ϕ k is a map from X k to Y . Following the terminology of [6,7], these are referred to as pseudo-polynomial functions.…”