Logic and Rational Languages of Scattered and Countable Series-Parallel Posets
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Let A be an alphabet and SP^(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^(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^(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^(A) is decidable.
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