On Rosser theories

Rosser theories play an important role in the study of the incompleteness phenomenon and meta-mathematics of arithmetic. In this paper, we first define the notions of $n$-Rosser theories, exact $n$-Rosser theories, effectively $n$-Rosser theories and effectively exact $n$-Rosser theories (see Definition 1.6). Our definitions are not restricted to arithmetic languages. Then we systematically examine properties of $n$-Rosser theories and relationships among them. Especially, we generalize some important theorems about Rosser theories for recursively enumerable sets in the literature to $n$-Rosser theories in a general setting.