A note on a property of maximal sets and choice in the absence of universal comparability

We show that, given a reflexive and transitive (but not necessarily connected) binary weak preference relation R, the maximal set generated by R coincides with the union of the choice sets generated by all possible orderings that are compatible with R. The property is of interest in assessing the intuitive significance of the maximal set for the choice of an agent whose preferences do not satisfy connectedness.