Extending an order on a set to the power set: Some remarks on Kannai and Peleg's approach
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 \(\{\) \(x\} \succ\{y\}\). Kannai and Peleg 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 {\#}\(\Omega \geq 6\) [see \textit{Y. Kannai} and \textit{B. Peleg}, ibid. 32, 172-175 (1984; Zbl 0533.90005)]. The authors commented on the Kannai-Peleg result by establishing two theorems. Their axioms are: (B) For all distinct x,\(y\in\Omega \), if xRy, then \(\{\) \(x\} \succ\{x,y\}\) and \(\{\) x,y)\(\succ\{y\}.\) (K) Let \(A,B\in 2^{\Omega}\backslash\{\Phi\}\). If [for all \(x\in A\) and for all \(y\in B\), xRy] and [for some \(x\in A\) and some \(y\in B\), xRy and not yRx], then \(A\succ B.\) (M') If \(A,B,C\in 2^{\Omega}\backslash\{\emptyset\}\), \(A\cap B=A\cap C=\emptyset\), and \(B\succ C\), then \(A\cup B\succ A\cup C.\) Axiom (K) is weaker than (GP) and axiom (B) is weaker than (K), while axiom (M') is stronger than M. The theorems are: (1) If \(\Omega\) is finite there exists a weak order over \(2^{\Omega}\backslash\{\emptyset\}\) satisfying (K) and (M). (2) If {\#}\(\Omega \geq 3\) there exists no binary relation \(\gtrsim\) on \(2^{\Omega}\backslash\{\emptyset\}\) which satisfies (B) and (M').
extension of a linear order to the power set
scientific article; zbMATH DE number 3874910