It is well known that the classic {\L}o\'s-Tarski preservation theorem fails in the finite: there are first-order definable classes of finite structures closed under extensions which are not definable (in the finite) in the existential fragment of first-order logic. We strengthen this by constructing for every $n$, first-order definable classes of finite structures closed under extensions which are not definable with $n$ quantifier alternations. The classes we construct are definable in the extension of Datalog with negation and indeed in the existential fragment of transitive-closure logic. This answers negatively an open question posed by Rosen and Weinstein.

中文翻译：

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一阶逻辑的有限类和前缀类中的外延保全

众所周知，经典的 {\L}o\'s-Tarski 保持定理在有限域中是失败的：存在一阶可定义的有限结构类在存在片段中不可定义（在有限域中）的扩展下封闭一阶逻辑。我们通过为每个 $n$ 构造在扩展下封闭的一阶可定义有限结构类来加强这一点，这些扩展不能用 $n$ 量词交替来定义。我们构造的类可以在带有否定的 Datalog 扩展中定义，实际上在传递闭包逻辑的存在片段中是可定义的。这否定了 Rosen 和 Weinstein 提出的一个悬而未决的问题。