A short note on essentially $\Sigma_1$ sentences (revised and extendend version)

Characterization of $\Sigma_1$-formulas with a binary operator interpreted as interpretability in finitely axiomatizable extensions of the theory $I\Delta_0+Superexp$ and some purposes concerning $I\Delta_0+exp$. In Provability Logic and in Interpretability Logic, More precisely: \begin{definition} (See [DJP]) Let $\mathbf{L}_0$ be a logic with a collection of arithmetical interpretations into an r.e. extension $\bT$ of ${\bf I\Delta_0+Supexp}$. We say that a formula $A$ is \emph{essentially} $\Sigma_1$ with respect to $\bT$, if $A^{\star}$ is provably equivalent to a $\Sigma_1$ formula for \emph{any} arithmetical interpretation $^\star$ of $\mathbf{L}_0$ in $\bT$.\end{definitionn} [DPJ] De Jongh and Pianigiani obtained some results relative to the system ${\bf R}$, with reference to any r.e. $\Sigma_1$-sound extension of ${\bf I \Sigma_1}$. In this note we show that a characterization of this kind can be obtained also for formulas of the logic ${\bf ILP}$ (cf [V1]), with respect to any finitely axiomatizable $\Sigma_1$-sound extension of ${\bf I\Delta_0 + Supexp}$.