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Logic and Rational Languages of Scattered and Countable Series-Parallel Posets
Let be an alphabet and denote the class of all countable
N-free partially ordered sets labeled by , in which chains are scattered
linear orderings and antichains are finite. We characterize the rational
languages of by means of logic. We define an extension of
monadic second-order logic by Presburger arithmetic, named P-MSO, such that a
language of is rational if and only if 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 is decidable