  A fundamental theme in automata theory is regular languages of words and
trees, and their many equivalent definitions. Salvati has proposed a
generalization to regular languages of simply typed $\lambda$-terms, defined
using denotational semantics in finite sets.
  We provide here some evidence for its robustness. First, we give an
equivalent syntactic characterization that naturally extends the seminal work
of Hillebrand and Kanellakis connecting regular languages of words and
syntactic $\lambda$-definability. Second, we show that any finitary extensional
model of the simply typed $\lambda$-calculus, when used in Salvati's
definition, recognizes exactly the same class of languages of $\lambda$-terms
as the category of finite sets does.
  The proofs of these two results rely on logical relations and can be seen as
instances of a more general construction of a categorical nature, inspired by
previous categorical accounts of logical relations using the gluing
construction.
