Hausdorr's Theorem for Posets That Satisfy the Nite Antichain Property

Hausdorr characterized the class of scattered linear orderings as the least family of linear orderings that includes the ordinals and is closed under ordinal summations and inversions. We formulate and prove a corresponding characterization of the class of scattered partial orderings that satisfy the nite antichain condition (FAC). Consider the least class of partial orderings containing the class of well-founded orderings that satisfy the FAC and is closed under the following operations: (1) inversion, (2) lexicographic sum, and (3) augmentation (where hP; i augments hP; i ii x y whenever x y). We show that this closure consists of all scattered posets satisfying the nite antichain condition. Our investigation also shed some light on the natural (Hessen-berg) sum of ordinals and the related product and exponentiation operations.