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:

inversion, lexicographic sum, and augmentation (where augments iff whenever ). We show that this closure consists of all scattered posets satisfying the finite antichain condition.

Our investigation also shed some light on the natural (Hessenberg) sum of ordinals and the related product and exponentiation operations.