We study FO+, a fragment of first-order logic on finite words, where monadic predicates can only appear positively. We show that there is an 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 lift this example language to finite graphs, thereby providing a new result of independent interest for FO-definable graph classes: negation might be needed even when the class is closed under addition of edges. We finally show that the problem of whether a given regular language of finite words is definable in FO+ is undecidable.

翻译：我们研究了FO+，即有限词上的一阶逻辑片段，其中一元谓词只能以肯定形式出现。我们证明存在一个在一元谓词下单调但不可用FO+定义的一阶可定义语言，这为林登保型定理在有限结构上失效提供了一个简洁的证明。我们将该示例语言推广至有限图，从而得到一个具有独立意义的新结果：对于一阶可定义的图类，即使该类在添加边关系下封闭，也可能需要否定操作。最后，我们证明给定正则有限词语言是否可用FO+定义的问题是不可判定的。