We consider arbitrary stratified languages. We study structures which satisfy the same stratified sentences and we obtain an extension of Keisler's Isomorphism Theorem to this situation. Then we consider operations which are definable by a stratified formula and modify the `type' of their argument by one; we prove that for such an operation F the sentence c = F(c) and the scheme $\varphi(c) \leftrightarrow \varphi(F(c))$ , where φ(x) varies among all the stratified formulas with no variable other than x free, (...) imply the same stratified {c}-sentences. (shrink)