Dimension de krull des ensembles ordonnes

The notion of deviation of an ordered set has been introduced by Gabriel as a tool to classify rings. It measures how far a given ordered set P deviates from ordered sets satisfying the descending chain condition. We consider here a more general notion and, according to Robson, we define the Krull dimension of P as the deviation of the collection F(P) of its final segments ordered by inclusion. Using a generalization of the partition theorem of Erd6s, Dushnik and Miller, due to Milner and Pouzet, we show that the Krull dimension of P is the maximum of the Krull dimension of its linear extensions. When P is partially well ordered (and the deviations is the usual one) this fact is an easy consequence of a result of de Jongh and Parikh asserting that P has a linear extension of maximum type (in fact it is an equivalent statement). Added to another result of these authors, it gives directly the value of the Krull dimension of a product of two partially well ordered sets, value previously computed by Robson.