The class of opportunity filters and the preference revealed by a path independent choice function

The purpose of this paper is to replicate the theory developed by Gekker (2001), without using any monotonicity assumption. We however retain a non-triviality assumption implicit in Gekker (2001), which says that there is at least one opportunity set which is preferred to the no-choice situation. In addition we require our preference relation on opportunity sets to be transitive (as in Gekker (2001), reflexive and satisfy an assumption called minimal comparability, which requires every opportunity set to be comparable with the null set. We also show that there exists a preference relation on the power set of the set of alternatives, revealed by a path independent choice function, which satisfies all the properties that we require of a binary relation to satisfy on the power set of the set of alternatives.