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 asymptotically e(-2) = 0.1353 .... (C) 1998 Academic Press.