Hausdorff ’s theorem for posets that satisfy the finite antichain property

Hausdorff 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 finite 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 ${\displaystyle ⟨P,⪯⟩}$ augments ⟨P, ≤⟩ iff ${\displaystyle x⪯y}$ whenever x ≤ y). We show that this closure consists of all scattered posets satisfying the