Prime two-dimensional orders and perpendicular total orders (Q1266382)

Two (partial) orders on the underlying set \(V\) are perpendicular if they do not share endomorphisms but the trivial ones (e.g., constant maps and the identity map). This notion was introduced by Demetrovics et al. The present paper gives a new proof for an earlier result of Nozaki et al.: Theorem. (1) Every total order on at least four elements has a perpendicular total order. (2) Given a total order \(\rho\) on a finite set, the proportion of the total orders which are perpendicular to \(\rho\) is asymptotically \(e^{-2}\). The paper also establishes new results on the asymptotic enumeration of 2-dimensional prime orders.