On the number of k-realizations of an ordered set

A k-realization of an ordered set P is a sequence of k linear orderings of the underlying set of P, whose intersection is (the order relation of) P. We determine the status of the number of k-realizations with respect to comparability invariance, and we show that among all orders on the set {1, 2, ..., n}, the antichain has the most k-realizations, for any k>1. The latter intuitively reasonable result rests ultimately on an observation related to comparability invariance for numbers of linear extensions.Research supported by ONR Contract N00014 85-K-0769.