On the threshold representation of partial orderings

In this paper, we provide a constructive proof for threshold representations of partial orderings in a finite setting and extend the construction to arbitrary sets. This contributes to a long-standing open problem posed in Nakamura (2002) and Aleskerov et al. (2007). We identify and discuss aspects of our proof that do not directly generalize to an infinite setting and, for each, provide intuitive sufficient conditions. • ≻ is a partial order if and only if it is metric representable (finite case). • Constructive proof via the minimum weighted path metric. • The representation is extended to infinite sets under mild conditions. • These resolve an open problem posed in Nakamura (2002) and Aleskerov et al. (2007).