Many-sorted elementary equivalence (Q1122567)

Though there is a close connection between many-sorted and one-sorted languages, there are some notions which are not directly transportable to the second case [\textit{J. L. Hook}, J. Symb. Logic 50, 372-374 (1985; Zbl 0571.03026)]. This paper is devoted to the study of four possible ways how to construct the one-sorted language corresponding to a many-sorted language. Special attention is given to the equivalence relation. The author studies conditions under which the suggested transition from the many-sorted to the one-sorted case has the following property: ``If \({\mathfrak M}\) and \({\mathfrak N}\) are equivalent structures of a many-sorted language, then the one-sorted structures \({\mathfrak M}^*\), \({\mathfrak N}^*\) canonically associated with them are equivalent, too.'' Special attention is given to simple type theory.