Fundamentals of planar ordered sets

Abstract A finite ordered set is planar when it can be represented in the plane with the point a lower than b in the plane whenever a is less than b in the ordered set, and with a straight-line segment from a to b whenever a is covered by b . We prove it is equivalent to allow instead an arc from a to b as long as the arc does not go below the level of a or above the level of b (whenever a is covered by b ). Our result is analogous to the well-known graph-theoretic result of K. Wagner and I. Fary.