Basically, the connection of two many-sorted theories is obtained by taking their disjoint union, and then connecting the two parts through connection functions that must behave like homomorphisms on the shared signature. We determine conditions under which decidability of the validity of universal formulae in the component theories transfers to their connection. In addition, we consider variants of the basic connection scheme. Our results can be seen as a generalization of the so-called ℰ-connection approach for combining modal logics to an algebraic setting.
"Connecting many-sorted theories." J. Symbolic Logic 72 (2) 535 - 583, June 2007. https://doi.org/10.2178/jsl/1185803623