  For simplicity, most of the literature introduces the concept of definitional
equivalence only to languages with disjoint signatures. In a recent paper,
Barrett and Halvorson introduce a straightforward generalization to languages
with non-disjoint signatures and they show that their generalization is not
equivalent to intertranslatability in general. In this paper, we show that
their generalization is not transitive and hence it is not an equivalence
relation. Then we introduce the Andr\'eka and N\'emeti generalization as one of
the many equivalent formulations for languages with disjoint signatures. We
show that the Andr\'eka-N\'emeti generalization is the smallest equivalence
relation containing the Barrett-Halvorson generalization and it is equivalent
to intertranslatability even for languages with non-disjoint signatures.
Finally, we investigate which definitions for definitional equivalences remain
equivalent when we generalize them for theories with non-disjoint signatures.
