  Let $A$ be an alphabet and $SP^\diamond(A)$ denote the class of all countable
N-free partially ordered sets labeled by $A$, in which chains are scattered
linear orderings and antichains are finite. We characterize the rational
languages of $SP^\diamond(A)$ by means of logic. We define an extension of
monadic second-order logic by Presburger arithmetic, named P-MSO, such that a
language $L$ of $SP^\diamond(A)$ is rational if and only if $L$ is the language
of a sentence of P-MSO, with effective constructions from one formalism to the
other. As a corollary, the P-MSO theory of $SP^\diamond(A)$ is decidable.
