Random orders

Letk andn be positive integers and fix a setS of cardinalityn; letP _k (n) be the (partial) order onS given by the intersection ofk randomly and independently chosen linear orders onS. We begin study of the basic parameters ofP _k (n) (e.g., height, width, number of extremal elements) for fixedk and largen. Our object is to illustrate some techniques for dealing with these random orders and to lay the groundwork for future research, hoping that they will be found to have useful properties not obtainable by known constructions.Supported by NSF grant MCS 84-02054.