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33 If f is a partial fuzzy voting rule, then f is a voting rule. , 1Rn ). Then (x, y) ∪ F P D (f ) (ρ) ∃ ⊃(σ, ω) ∪ D(f ), ∩i ∪ Supp(σ ), 1Pi (x, y) = 1, ∩j ∪ Supp(ω), 1Rj (x, y) = 1 ∃ ⊃(σ, ω) ∪ D(f ), ∩i ∪ Supp(1Supp(σ ) ), 1Pi (x, y) = 1, ∩j ∪ Supp(1Supp(ω) ), 1Rj (x, y) = 1 ∃ ⊃S, W ∪ D(f ), ∩i ∪ S, (x, y) ∪ Pi , ∩j ∪ W , (x, y) ∪ Rj , where S = Supp(1Supp (σ )), W = Supp(1Supp(ω) ) for the implication ⇒ and σ = 1S and ω = 1W for the implication ⇐. , Rn ). Thus f = fD (f ) . 34 A fuzzy aggregation rule is a partial fuzzy voting rule if and only if it is neutral and monotonic.

3 Fuzzy Voting Rules 43 or πi (y, z) > 0∩i ∪ Supp(σ ). Thus it follows that F P D (ρ) = {(x, y)}, Symm(F P D (ρ)) = {(x, y), (y, x)} and f (ρ)(x, y) = {πi (x, y) ∧ ρj (x, y) | i = 1, 2; j = 1, 2, 3} 3 3 3 1 3 1 1 3 1 1 1 1 { ∧ , ∧ , ∧ , ∧ , ∧ , ∧ } 4 4 4 2 4 8 2 4 2 2 2 8 = = 3 . 4 It also follows that fD (ρ)(u, v) = 1 if (y, x) ⊂= (u, v) ⊂= (x, y) and that f (y, x) = 0. 32 Let f be a fuzzy aggregation rule. Then f is called a partial fuzzy voting rule if ∩ρ ∪ F R n and ∩x, y ∪ X, f (ρ)(x, y) > 0 if and only if fD (f ) (ρ)(x, y) > 0.

2) If f is weakly Paretian, then f is weakly Paretian. , 1P (xm−1 , xm ) > 0 implies 1R (x1 , xm ) > 0, where P and R correspond to f . , n. (2) Since f is weakly Paretian, ∩ρ ∪ R n , it follows that ∩x, y ∪ X, (∩i ∪ N, 1Pi (x, y) > 0 implies 1P (x, y) > 0). Thus ∩i ∪ N, xPi y implies xPy. The following result states in words that any fuzzy preference aggregation rule that is partially acyclic and weakly Paretian results in a fuzzy collegium. 20 Let f be a fuzzy aggregation rule. If f is partially acyclic and weakly Paretian, then f is collegial.

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