  One of the main reasons for the correspondence of regular languages and
monadic second-order logic is that the class of regular languages is closed
under images of surjective letter-to-letter homomorphisms. This closure
property holds for structures such as finite words, finite trees, infinite
words, infinite trees, elements of the free group, etc. Such structures can be
modelled using monads. In this paper, we study which structures (understood via
monads in the category of sets) are such that the class of regular languages
(i.e. languages recognized by finite algebras) are closed under direct images
of surjective letter-to-letter homomorphisms.
  We provide diverse sufficient conditions for a monad to satisfy this
property. We also present numerous examples of monads, including positive
examples that do not satisfy our sufficient conditions, and counterexamples
where the closure property fails.
