A uniqueness result for extending orders; with application to collective choice as inconsistency resolution (Q791414)

Given a reflexive linear order R on a set \(\Omega\), and hence given its asymptotic part P, a weak order \(\gtrsim\) on the power set \(2^{\Omega}\) is called an extension of P, if for all x, \(y\in \Omega\) for which xPy, it follows that \(\{x\}\succ\{y\}\). \textit{Y. Kannai} and \textit{B. Peleg} [ibid. 32, 172-175 (1984)] proposed two appealing axioms for such extensions, (GP) and (M), and established their impossibility theorem that there is no extension of a reflexive linear order P on \(\Omega\) to \(2^{\Omega}\backslash \{\emptyset \}\) which satisfies both (GP) and (M) and is also a weak ordering, if {\#}\(\Omega\geq 6.\) The authors commented on the Kannai-Peleg result by quoting their possibility theorem. Their axioms \(are:\) \((GP^{**})\) If \(A,B\in 2^{\Omega}\), \(A\neq \emptyset\) and xPy for all \(x\in A\) and \(y\in B\) then \(A\succ B;\) \((M^{**})\) If \(A,B,C\in 2^{\Omega}\), \(A\cap B=\emptyset =A\cap C\), then \(B\gtrsim C\) iff \(A\cup B\gtrsim A\cup C.\) The Heiner-Packard theorem says: Suppose R is a weak ordering on a finite set \(\Omega\) and that \(\gtrsim\) is an extension of R onto \(2^{\Omega}\). Then \(\gtrsim\) is a weak ordering on \(2^{\Omega}\) and satisfies \((GP^{**})\) and \((M^{**})\), iff \(\gtrsim\) is the lexicographic ordering according to the number of highest-ranked elements in \(\Omega\) ordered by R.