Voting by Committees under Constraints

A section is a group of objects with the property that the decision among
their active components can be made without paying attention to the infea-
sibilities involving objects on its complement.

Definition 9 A subset of objects B Ç K is a section of Rf if for all active
components B', B" ∈ AC (B) we have Cf (B') = Cf(B").

Remark 1 Rf is a section of Rf because Cf (A) is empty for all active
components X ∈ AC (Rf) = Rf-

Given two families of subsets of objects X and У we denote by X + У
the sum of the two; namely,

X + У = {A U Y ∈ 2^κ I X ∈ XandY ∈ J}.

Remark 2 B is a section of Rf if and only if, for all B' ∈ ΛC(B),

R_f = AC(B) R Cf (B^,).

Lemma 1 Let B be a section of Rf and let B₁ and B₂ be such that
B = Bi U B₂, B₁ ∩ B₂ = 0, and B_t is a section of Rf- Then, B₂ is also a
section of R_f .

Proof By definition of active component of B₂, for any X, Y ∈ R_f ,

X₂ ≡ X ∩ B₂ ∈ AC(B₂)                   (1)

and

Y₂ ≡ Y ∩ B₂ ∈ AC(B₂).

Moreover, by definition of range complement of A₂ and Y₂ relative to B₂,

XnB^c₂ECf²(X₂)

and

YnB^cECf²(Y₂).

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