Fragments of $\mathsf{I}\mathsf{O}\mathsf{p}\mathsf{e}\mathsf{n}$

In this paper we consider some fragments of $\mathsf{I}\mathsf{O}\mathsf{p}\mathsf{e}\mathsf{n}$ (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 $\mathsf{I}(\mathsf{l}\mathsf{i}\mathsf{t})$ is equivalent to $\mathsf{I}\mathsf{O}\mathsf{p}\mathsf{e}\mathsf{n}$ and is not finitely axiomatizable over $\mathsf{Q}$, establish some inclusion relations between $\mathsf{I}(=),\mathsf{I}(\mathsf{\ne }),\mathsf{I}(⩽)$ and $\mathsf{I}(\nleqq )$. We also prove that the set of diophantine equations solvable in models of $\mathsf{I}(=)$ is (algorithmically) decidable.