In this paper we consider some fragments of $$\textsf{IOpen}$$ (Robinson arithmetic $$\mathsf Q$$ with induction for quantifier-free formulas) proposed by Harvey Friedman and answer some questions he asked about these theories. We prove that $$\mathsf {I(lit)}$$ is equivalent to $$\textsf{IOpen}$$ and is not finitely axiomatizable over $$\mathsf Q$$, establish some inclusion relations between $$\mathsf {I(=)}, \mathsf {I(\ne )}, \mathsf {I(\leqslant )}$$ and $$\textsf{I} (\nleqslant )$$. We also prove that the set of diophantine equations solvable in models of $$\mathsf I (=)$$ (...) is (algorithmically) decidable. (shrink)