The PT-order,minimal cutsets and menger property

If P is a poset, the associated PT-order is the quasi order in which a b holds if every maximal chain of P which passes through a also passes through b. P is special if whenever A is a chain in P and a=sup A or inf A, then there is b A such that b a. It is proved that if P is chain complete and special then the set of -maximal elements is -dominating and contains a minimal cutset. As corollaries of this, we give partial answers to (i) a question of Rival and Zaguia by showing that if P is regular and special every element is in a minimal cutset and (ii) a question of Brochet and Pouzet by showing that if P is chain complete and special then it has the Menger property.Research partially supported by grants from the National Natural Science Foundation of China and the Natural Science Foundation of Shaanxi province