Prime Two-dimensional Orders and Perpendicular Total Orders

Starting with a correspondence between prime two-dimensional orders and pairs of perpendicular total orders we put in perspective several asymptotic results, we deduce an estimate of the number of prime two-dimensional orders (labelled and unlabelled as well). Using Poisson approximation, we give a new proof of the fact that the proportion of total orders perpendicular to a given total order is asymptoticallye−2 = 0.1353 ….