How to construct a partition when preference sets are given (Q1342942)

Let \(E\) and \(I\) be two sets. A preference set family for \(E\) is a family \((A_{ij})_{i,j\in I}\) of subsets of \(E\) such that \(A_{ii}= E\) for every \(i\in I\) and \(A_{ij}= E\backslash A_{ji}\) for every \(i,j\in I\) with \(i\neq j\). The relation \(x\in A_{ij}\) is interpreted as \(x\) prefers \(i\) to \(j\). The main result in the paper states that, if the transitivity property \(A_{ij}\cap A_{jk}\subset A_{ik}\), for every \(i,j,k\in I\) holds, then the family \((\bigcap_{j\in I} A_{ij})_{i\in I}\) is a partition of \(E\).