Monadic MSO logic

The correspondence of regular languages and monadic second-order logic relies on the fact that the class of regular languages is closed under images of surjective letter-to-letter homomorphisms. This closure property holds for finite words, but also for infinite words, finite or infinite trees, and more generally for a wide range of monads. However, it does not hold always, and a structural characterization of those monads for which it does hold seems elusive. I will focus on a few examples to illustrate how subtle the condition can be.