We study FO+, a fragment of first-order logic on finite words, where monadic predicates can only appear positively. We show that there is a FO-definable language that is monotone in monadic predicates but not definable in FO+. This provides a simple proof that Lyndon's preservation theorem fails on finite structures. We additionally show that given a regular language, it is undecidable whether it is definable in FO+.

中文翻译：

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单词的正一阶逻辑

我们研究FO +，这是有限词上一阶逻辑的一部分，其中一元谓词只能出现正数。我们表明，有一种FO可定义的语言，该语言在单子谓词中是单调的，但在FO +中是不可定义的。这提供了林登保留定理在有限结构上失败的简单证明。我们还表明，给定一种常规语言，它是否可以在FO +中定义是不确定的。