On categoricity of linear orders

We consider different notions of categoricity of linear orders. Recall that, a computable algebraic structure is called $\Delta^0_\alpha$-categorical if for any two computable copies of it there exists a $\Delta^0_\alpha$-isomorphism between them. A computable algebraic structure is called relatively $\Delta^0_\alpha$-categorical if for any two $x$-computable copies of it there exists a $\Delta^x_\alpha$-isomorphism between them. S. S. Goncharov and V. D. Dzgoev [2] and, independently, J. B. Remmel [5] gave a characterization of computably categorical linear orders. They proved that a computable linear order is computably categorical iff it has finitely many successors. C. McCoy [4] gave a characterization of relatively $\Delta^0_2$ categorical linear orders. He proved, that if a linear order $\mathcal{L}$ has a computable copy with a computable successor relation, and computable left and right limit points then $\mathcal{L}$ is $\Delta^0_2$ categorical iff it is relatively $\Delta^0_2$ categorical. C. Ash [1] found levels of categoricity of constructive ordinals. Namely, C. Ash proved that if an ordinal $\alpha$ such that $\omega^{\delta+n}\leq\alpha< \omega^{\delta+n+1}$ then $\alpha$ is $\Delta^0_{\delta+2n}$ categorical, and is not $\Delta^0_{\beta}$ categorical for any $\beta<\delta+2n$.
In the first part of talk we consider categoricity of scattered linear orders such that ranks of them are constructive ordinals. In the second part of talk we consider bi-embeddable categoricity of scattered linear orders with finite rank. In both cases we give upper and lower bounds of categoricity levels of considered linear orders.

Список литературы

1. C. J. Ash, “Recursive labelling systems and stability of recursive structures in hyperarithmetical degrees”, Transactions of the American Mathematical Society, 298:2 (1986), 497–514
2. С. С. Гончаров, В. Д. Дзгоев, “Автоустойчивость моделей”, Алгебра и логика, 19:1 (1980), 45–58    
3. Ч. Ф. Мак-Кой, “О $\Delta_3^0$-категоричности для линейных порядков и булевых алгебр”, Алгебра и логика, 41:5 (2002), 531–552      ; Algebra and Logic, 41:5 (2002), 295–305    
4. C. F. D. McCoy, “$\Delta^0_2$-categoricity in Boolean algebras and linear orderings”, Annals of Pure and Applied Logic, 119:1-3 (2003), 85–120
5. J. B. Remmel, “Recursive categorical linear orderings”, Proc. Am. Math. Soc, 83:2 (1981), 387–391