Prime Two-dimensional Orders and Perpendicular Total Orders

Abstract

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 . . ..

Weak orders admitting a perpendicular linear order

“…Theorem 2 gathers several results proved independently. Equivalence (i) ⇔ (v) is due to Rival and Zaguia [9], equivalence (ii) ⇔ (iii) to Nozaki et al [7] and equivalence (iii) ⇔ (iv) to the second author of the present paper [11]. For a direct proof of Theorem 2, obtain the equivalence (i) ⇔ (ii) from the fact that P P ∩ Q Q = L L ∩ L L ; next, use or prove the equivalences (ii) ⇔ (iii) and (iii) ⇔ (iv) and observe that the implications (iv) ⇒ (v) ⇒ (iii) are trivial (Fig.…”

“…Theorem 1 is due to Nozaki et al [7] (see [11] for a new proof, based on a probabilistic argument). Theorem 2 gathers several results proved independently.…”

Pairs of orthogonal countable ordinals

“…Theorem 4 was obtained for finite posets in [12], and extended to the infinite in [15]. Theorem 5 was stated in [25] for a finite sets; the proof given holds without that restriction.…”

“…With this terminology, the above fact can be expressed by saying that two linear orders L and M on the same finite set V are orthogonal if and only if the binary structure B := (V, L, M), that we call a bichain, is prime. This leads to results relating primality and orthogonality ( [22], [25]).…”

Profile and hereditary classes of ordered relational structures

“…These bichains are critical. Indeed, a bichain is indecomposable if and only if the intersection order is indecomposable ( [30] for finite bichains and [33] for infinite bichains). The isomorphic types of these bichains are described in Albert and Atkinson's paper in terms of permutations of 1, .…”