Morphic characterizations with insertion systems controlled by a context of length one

Among the variety of representation theorems for context-free languages, the Chomsky–Schützenberger theorem is unique in that it consists of Dyck languages, regular languages, and simple operations. In the present paper, we obtain some characterizations of and representation theorems for languages in the Chomsky hierarchy by using insertion systems, strictly locally testable languages, and morphisms in the framework of the Chomsky–Schützenberger theorem. For instance, each context-free language L can be represented in the form L=h(L(γ)∩R), where γ is an insertion system of weight (1,1), R is a strictly 2-testable language, and h is a morphism. In contrast, if R is instead a strictly 1-testable language, the class of languages generated is a proper subclass of context-free languages and incomparable with the class of regular languages.