On automatic and decidable linear orderings

Let $\EuScript A$ be the class of automatic linear orderings, $\EuScript{AA}$ be the class of linear orderings which are computably categorical in the class of automatic presentation, $\EuScript{AD}$ be the class of linear orderings which are computably categorical in the class of decidable presentations. Obviously, $\EuScript{AD}\cap\EuScript A\subset\EuScript{AD}$. Since all automatic structures are decidable, $\EuScript{AD}\cap\EuScript A\subset\EuScript{AA}$, and one can easily see that $\EuScript{AD}\cap\EuScript A$ is nonempty. We show that there exist a linear order $\mathcal L_1\in\EuScript{AA}$ such that $\mathcal L_1\notin\EuScript{AD}$ and a linear order $\mathcal L_2\in\EuScript{AD}$ such that $\mathcal L_2\notin\EuScript A$. By this, the inclusions $\EuScript{AD}\cap\EuScript A\subset\EuScript{AD}$ and $\EuScript{AD}\cap\EuScript A\subset\EuScript{AA}$ are proper. In addition, we construct an example of a non–automatic linear order which is decidable in the language with the additional quantifier $\exists^\infty$.