On Generalization of Definitional Equivalence to Non-Disjoint Languages

For simplicity, most of the literature introduces the concept of definitional equivalence only for disjoint languages. In a recent paper, Barrett
and Halvorson introduce a straightforward generalization to non-disjoint
languages and they show that their generalization is not equivalent to in-
tertranslatability in general. In this paper, we show that their generalization
is not transitive and hence it is not an equivalence relation. Then we intro-
duce another formalization of definitional equivalence due to Andréka and
Németi which is equivalent to the Barrett–Halvorson generalization in the
case of disjoint languages. We show that the Andréka–Németi generaliza-
tion is the smallest equivalence relation containing the Barrett–Halvorson
generalization and it is equivalent to intertranslatability, which is another
definition for definitional equivalence, even for non-disjoint languages. Finally, we investigate which definitions for definitional equivalences remain
equivalent when we generalize them for theories in non-disjoint languages.