On Extensions, Linear Extensions, Upsets and Downsets of Ordered Sets (Extended Abstract)

Let P be an order on a set of elements E We describe two distinct characterizations of the set of extensions of P. The first characterization is a one-to-one correspondence between extensions of P and pairs of upsets and downsets of certain suborders of P. The other characterizes the extensions of P in terms of linear extensions and sequences of upsets. This latter characterization leads to an algorithm that generates all the extensions of P in amortized time O(∣E∣2) per extension.