Ordinal-indexed sequences My question regards, as the title implies, ordinal-indexed sequences as it seems to be rather hard to find definite details about them.
At first, can I imagine an ordinal-indexed sequence as the following?:
Let $\alpha$ be an ordinal number(0,1,...,$\omega$,$\omega +1$,...). Then an ordinal indexed sequence is simply a function from the associated set of this ordinal to the specific alphabet $\Sigma$, i.e. taking the ordinal as a set of its predecessors, we have $s:\alpha\to\Sigma$.
From this consideration, the would be sequences possible like $010010001...2$ with an infinite sequence $010010001...$ indexed by all finite ordinals and another symbol $2$ at the end indexed by $\omega$. The ordinal of this sequence would then be $\omega +1$.
My second question is more of soft one, as I'm curious about the current research state of these type of sequences and what introduction literature is maybe out there.
 A: Your idea has not only been considered for ordinals but also for linear orders. The research initially focused on extending Kleene's theorem (equivalence between finite automata and regular expressions) and on logical characterizations. For words indexed by the ordinal $\omega$, the theory is mainly due to Büchi [2]. Early works also include [3,4,6], but I strongly recommend the article [1] which treats your question in some detail and contains an extensive bibliography. The book [5] is the reference book on linear orders.
[1] V. Bruyère and O. Carton, Automata on linear orderings, J. Comput. System Sci. 73, 1, 1-24, 2007.
[2] J. R. Büchi. On a decision method in the restricted second-order arithmetic. In Proc. Int. Congress Logic, Methodology and Philosophy of science, Berkeley 1960, pages 1–11. Stanford University Press, 1962.
[3] J. R. Büchi. Transfinite automata recursions and weak second order theory of ordinals. In Proc. Int. Congress Logic, Methodology, and Philosophy of Science, Jerusalem 1964, pages 2–23. North Holland, 1965.
[4] Y. Choueka. Finite automata, definable sets, and regular expressions over ωn-tapes. J. Comput. System Sci., 17(1):81–97, 1978.
[5] J. G. Rosenstein. Linear Orderings. Academic Press, New York, 1982.
[6] J. Wojciechowski. Finite automata on transfinite sequences and regular expressions. Fundamenta informaticæ, 8(3-4):379–396, 1985.
