Logic and rational languages of scattered and countable series-parallel posets.

Let A be an alphabet and S P ⋄ (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 S P ⋄ (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 S P ⋄ (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 S P ⋄ (A) is decidable. [ABSTRACT FROM AUTHOR]