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：我们研究有限词上的一阶逻辑片段FO+，其中一元谓词仅能以正形式出现。我们证明存在一个在一元谓词上单调但无法在FO+中定义的一阶可定义语言。这为林登定理在有限结构上不成立提供了简洁证明。我们将该示例语言推广至有限图，从而为图论中一阶可定义类带来全新结论：即便图类对边添加封闭时，否定算子仍可能是必需的。最终我们证明，判定有限词的给定正则语言是否可在FO+中定义的问题是不可判定的。