Linear extensions of a partial order and NaP preferences

The conditions of completeness and transitivity and their violations are important in theories of preferential choice. NaP (necessary and possible) preferences try to disentangle their interplay. They are related to so called Richter Peleg representations of partial orders, where the order is represented by a vector of numerical functions instead of a single one. They consist of splitting a quasiorder (preorder) into two nested relations. Recently, a series of papers (e.g. Gialotta & Watson (2018, 2020)) gave generalizations of this concept. It turns out that these results are rather straightforward consequences of (generalizations of) Szpilrajn’s theorem on the linear extensions of a partial order. In the present paper we investigate the set of linear extensions in order to clarify what is behind NaP preferences and their generalizations. In doing so a natural extension on probabilistic NaP preferences is suggested.