There are correspondences between regular languages and finite automata, and $\omega$-regular languages and Buchi or Muller automata (as well as the characterisation in terms of the monadic second order logic of $\omega$).
But what about if we take words indexed by some or other fixed order type which is neither finite, nor equal to $\omega$, nor equal to $\omega^*$? For example take words to just be elements of $\Sigma^{\alpha}$ where $\alpha$ is some fixed linear order, say $\Sigma^{\omega+\omega^*}$ or $\Sigma^{\mathbb{Z}}$, with $\Sigma$ a finite alphabet.
If there is something that makes sense only for discrete linear orders $\alpha$ this would be fine, but ideally I am after something that works not just for well-orders (I am aware of Buchi's work on `transfinite automata recursions' in this direction).
Any references to literature would be greatly appreciated.
The answer is yes. There are several papers on this topic, see the references of [1, 2, 4] for an extensive bibliography. Note also the extension to tree automata in [3].
[1] N. Bedon, A. Bès, O. Carton, and C. Rispal, Logic and rational languages of words indexed by linear orderings, Theory Comput. Syst. 46 (2010), no. 4, 737--760.
[2] V. Bruyère and O. Carton, Automata on linear orderings, J. Comput. System Sci., 73(1):1–24, (2007).
[3] V. Bruyère, O. Carton, and G. Sénizergues, Tree automata and automata on linear orderings, Theoret. Informatics and Applications 43, no. 2, pp. 321-338, 2009.
[4] O. Carton, T. Colcombet, G. Puppis. Regular languages of words over countable linear orderings, Automata, languages and programming. Part II, 125--136, LNCS 6756, Springer, Heidelberg, 2011.