Resumen de Ordering the set of antichains of an ordered set

Paulo Ribenboim

• The purpose of this paper is to define, in a natural way, an order relation on the set of antichains of an ordered set. Thus, the set of finite antichains becomes a sup-lattice. If moreover, the given ordered set is a strictly ordered commutative monoid, the finite antichains constitute also an ordered monoid.\newline In $\S$ 3, an ultrametric distance is introduced; it has values in the partially ordered set of finite antichains. This distance appears naturally in the study of rings of generalized power series. It will be used in a forthcoming paper (with Sibylla Prie$\beta$-Crampe) concerning fixed points of contracting maps. Antichains have also been considered in the study of Gröbner bases, where they appear under names like "crowns" and it is likely that the present results will be useful in that context.\newline By its simplicity, it is also expected that use of these ideas will be found in the theory of ordered sets and their applications.