Cette étude s'inscrit dans un prolongement algorithmique d'un travail de Bruno Leclerc, publié dans cette revue, qui discute de la taille maximum d'une antichaîne dans un produit direct P d'ordres totaux. On y présente un algorithme de partitionnement de P en un nombre minimum de chaînes. Enfin, on décrit brièvement une application à l'extraction de connaissance.

We define new combinatorial objects, called shrubs, such that forests of rooted trees are shrubs. We then introduce a structure of operad on shrubs. We show that this operad is contained in the Zinbiel operad, by using the inclusion of Zinbiel in the operad of moulds. We also prove that this inclusion is compatible with the richer structure of anticyclic operad that exists on Zinbiel and on moulds.

A separoid is a symmetric relation $†\subset \left(\genfrac{}{}{0pt}{}{2^S}{2}\right)$ defined on disjoint pairs of subsets of a given set $S$ such that it is closed as a filter in the canonical partial order induced by the inclusion (i.e., $A†B⪯A^{\text{'}}†B^{\text{'}}\iff A\subseteq A^{\text{'}}$ and $B\subseteq B^{\text{'}}$). We introduce the notion of homomorphism as a map which preserve the so-called “minimal Radon partitions” and show that separoids, endowed with these maps, admits an embedding from the category of all finite graphs. This proves that separoids constitute a countable universal partial order. Furthermore,...