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We show that an extension of MSO with Presburger arithmetic, named P-MSO, is as expressive as branching automata over scattered and countable N-free posets. As a consequence of the effectiveness of the constructions from one formalism to the other, the P-MSO theory of the scattered and countable N-free posets is decidable.

Logic and Rational Languages of Scattered and Countable Series-Parallel Posets