Omega-自动结构上的基数和计数量词
Cardinality and counting quantifiers on omega-automatic structures
摘要：
本文研究了可以用ω-自动机表示的结构，即所谓的ω-自动结构，证明了在一阶逻辑中由一阶量词“至多存在$\aleph_0$多”、"存在多个“和”存在$k$模$m$多“展开的这种结构上定义的关系是ω-正则的.证明确定某些代数性质的欧米加半群。因此，可数指标的欧米伽正则等价关系具有欧米伽正则代表集。这意味着Blumensath的猜想，一个可数的结构与$ω$ -自动表示可以表示使用自动机上有限的话。这也补充了最近的结果Hjörth，Khoussainov，Montalban和Nies表明，有一个欧米茄自动结构，没有内射介绍。
英文摘要：
We investigate structures that can be represented by omega-automata, so called omega-automatic structures, and prove that relations defined over such structures in first-order logic expanded by the first-order quantifiers `there exist at most $\aleph_0$ many', 'there exist finitely many' and 'there exist $k$ modulo $m$ many' are omega-regular. The proof identifies certain algebraic properties of omega-semigroups. As a consequence an omega-regular equivalence relation of countable index has an omega-regular set of representatives. This implies Blumensath's conjecture that a countable structure with an $ω$-automatic presentation can be represented using automata on finite words. This also complements a very recent result of Hjörth, Khoussainov, Montalban and Nies showing that there is an omega-automatic structure which has no injective presentation.
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