Dimension de Krull des ensembles ordonnés. (Krull dimension of ordered sets) (Q1066924)

The notion of deviation of a partially ordered set has been introduced by Rentschler and Gabriel in 1967, but only for finite ordinals. This definition was extended to arbitrary ordinals by \textit{B. Lemonnier} [Bull. Sci. Math., II. Sér. 96, 289-303 (1972; Zbl 0258.06002)]. Curiously, no mention about this paper is given in the paper under review. The aim of this paper is to prove the following result: let P be a poset and \(\Gamma\) a set of countable and ''impartible'' ordinals; then, the \(\Gamma\)-Krull dimension of P is the maximum of the \(\Gamma\)-Krull dimensions of its linear extensions. The \(\Gamma\)-Krull dimension of P is by definition (following Robson) the \(\Gamma\)-deviation of the set of its final segments ordered by inclusion; the notion of \(\Gamma\)-deviation of a poset X is an extension of the notion of deviation of X, measuring how much is deviated X from the class of posets not containing any ordinal type of \(\Gamma\). The paper contains in its final section some applications.