Positive First-order Logic on Words

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 additionally show that given a regular language, it is undecidable whether it is definable in FO + .