The infinite random order of dimension k. (English) Zbl 0589.06001

Winkler, Peter

Publ. Dép. Math., Nouv. Sér., Univ. Claude Bernard, Lyon 2/B, 37-40 (1985).

Fix an integer \(\kappa\geq 1\) and let \(I^{\kappa}\) be the unit hypercube in Euclidean \(\kappa\)-space, endowed with the ordinary product order. If n points are chosen randomly and independently from the uniform probability distribution on \(I^{\kappa}\), the resulting ordered set is called the random order \(P_ n^{\kappa}\). The author discusses the problem, whether ”the 0-1 law” holds for random orders, i.e. a theorem saying that for any first order sentence S in the language of ordered sets the \(\lim_{n\to \infty}P(S\) holds in \(P_ n^{\kappa})\) is either 0 or 1. The conjecture is made that the answer is yes in general, no for fixed or bounded dimension.

Reviewer: V.N.Salij

MSC:

06A06	Partial orders, general
03C90	Nonclassical models (Boolean-valued, sheaf, etc.)
60C05	Combinatorial probability

Keywords:

product order; uniform probability; random orders; first order sentence; language of ordered sets