  This article extends a paper of Abraham and Bonnet which generalised the
famous Hausdorff characterisation of the class of scattered linear orders.
Abraham and Bonnet gave a poset hierarchy that characterised the class of
scattered posets which do not have infinite antichains (abbreviated FAC for
finite antichain condition). An antichain here is taken in the sense of
incomparability. We define a larger poset hierarchy than that of Abraham and
Bonnet, to include a broader class of ``scattered'' posets that we call
$\kappa$-scattered. These posets cannot embed any order such that for every two
subsets of size $ < \kappa$, one being strictly less than the other, there is
an element in between. If a linear order has this property and has size
$\kappa$ we call this set $\qkappa$. Such a set only exists when
$\kappa^{<\kappa}=\kappa$. Partial orders with the property that for every
$a<b$ the set $\{x: a<x<b\}$ has size $\geq \kappa$ are called weakly
$\kappa$-dense, and partial orders that do not have a weakly $\kappa$-dense
subset are called strongly $\kappa$-scattered. We prove that our hierarchy
includes all strongly $\kappa$-scattered FAC posets, and that the hierarchy is
included in the class of all FAC $\kappa$-scattered posets. In addition, we
prove that our hierarchy is in fact the closure of the class of all
$\kappa$-well-founded linear orders under inversions, lexicographic sums and
FAC weakenings. For $\kappa=\aleph_0$ our hierarchy agrees with the one from
the Abraham-Bonnet theorem.
