Let A be an alphabet and $S{P}^{\diamond }(A)$ 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 $S{P}^{\diamond }(A)$ 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 $S{P}^{\diamond }(A)$ 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 $S{P}^{\diamond }(A)$ is decidable.

中文翻译：

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分散且可数的串并行摆态的逻辑和有理语言

设A为字母，$\mathrm{小号}{P}^{\diamond }（\mathrm{一种}）$表示由A标记的所有可计数的无N部分偏序集的类别，其中链是分散的线性排序，而反链是有限的。我们刻画了$\mathrm{小号}{P}^{\diamond }（\mathrm{一种}）$通过逻辑。我们通过Presburger算术定义了单子二阶逻辑的扩展，称为P-MSO，使得语言L为$\mathrm{小号}{P}^{\diamond }（\mathrm{一种}）$当且仅当L是P-MSO句子的语言，并且具有从一种形式主义到另一种形式主义的有效构造时，它才是合理的。因此，P-MSO理论$\mathrm{小号}{P}^{\diamond }（\mathrm{一种}）$ 是可以决定的。