可数序数上的一阶分离性
First-order separation over countable ordinals
摘要：
证明了可数序数词上分离两个一元二阶公式的一阶公式的存在性是可判定的。这扩展了Henckell和Almeida关于有限词的工作，以及Place和Zeitoun关于$ω$ -词的工作。为此，我们发展了么半群的代数概念（分别为）。$ω$ -半群序幺半群）与非周期合并，一个扩展幺半群（分别. $ω$ -半群序数幺半群），其显式地包括捕获由一阶不可逆性引起的精度损失的新操作。我们还展示了FO-点状集的可计算性，以及可数序数词上一阶逻辑的覆盖问题的可判定性。
英文摘要：
We show that the existence of a first-order formula separating two monadic second order formulas over countable ordinal words is decidable. This extends the work of Henckell and Almeida on finite words, and of Place and Zeitoun on $ω$-words. For this, we develop the algebraic concept of monoid (resp. $ω$-semigroup, resp. ordinal monoid) with aperiodic merge, an extension of monoids (resp. $ω$-semigroup, resp. ordinal monoid) that explicitly includes a new operation capturing the loss of precision induced by first-order indistinguishability. We also show the computability of FO-pointlike sets, and the decidability of the covering problem for first-order logic on countable ordinal words.
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