  The notion of level posets is introduced. This class of infinite posets has
the property that between every two adjacent ranks the same bipartite graph
occurs. When the adjacency matrix is indecomposable, we determine the length of
the longest interval one needs to check to verify Eulerianness. Furthermore, we
show that every level Eulerian poset associated to an indecomposable matrix has
even order. A condition for verifying shellability is introduced and is
automated using the algebra of walks. Applying the Skolem--Mahler--Lech
theorem, the ${\bf ab}$-series of a level poset is shown to be a rational
generating function in the non-commutative variables ${\bf a}$ and ${\bf b}$.
In the case the poset is also Eulerian, the analogous result holds for the
${\bf cd}$-series. Using coalgebraic techniques a method is developed to
recognize the ${\bf cd}$-series matrix of a level Eulerian poset.
