Choice functions over a finite set: A summary

A choice function picks some outcome(s) from every issue (subset of a fixed set A of outcomes). When is this function derived from one preference relation on A (the choice set being then made up of the best preferred outcomes within the issue), or from several preference relations (the choice set being then the Pareto optimal outcome within the issue, or the union of the best preferred outcomes for each preference relation)? A complete and unified treatment of these problems is given based on three functional properties of the choice function. None of the main results is original.