Elementary Theory of Ordinals with Addition and Left Translation by omega

After Büchi it has become very natural to interprete formulae of certain logical theories as finite automata, i.e., as recognizing devices. This recognition aspect though, was neglected by the inventor of the concept and the study of the families of linear structures that could be accepted in the language theory sense of the term, was carried out by other authors. The most popular field of application of Büchi type automata is nowadays connected with model checking by considering a process as a possibly infinite sequence of events. For over a decade, the original model has been enriched by adding a time parameter in order to model reac-tive systems and their properties. Originally Büchi was interested in the monadic second order theory with the successor over ω but he later considered the theory of countable ordinals for which he was led to propose new notions of finite automata. Again these constructs can be viewed as recognizing devices of words over a finite alphabet whose length are countable ordinals. They were studied by other authors, mainly Choueka and Wojciechowski to who we owe two Theorems " ` a la Kleene " asserting the equivalence between expressions using suitable rational operators and subsets (languages) of transfinite words, [6] and [13]. Lately, there has been a renewed interest for strings of transfinite lengths as such by shifting the emphasis from logic to language theory. This is testified by various attempts to extend results from finite to transfinite strings. E.g., Eilenberg's famous varieties theorem suggested by Schützenberger's characterization of star-free languages, asserts that there exists a " natural " bijection between certain families of languages (i.e., subsets of finite length words) and certain families of finite semi-groups. In both cases these families are defined by simple closure properties , [8]. In order to extend this result to words of transfinite length, Bedon and Carton extended Wilke's ω-semigroups to so-called ω 1-semigroups and were able that Eilenberg's theorem extends naturally, [1]. Also, the theory of rational relations which studies the rational subsets of pairs of words was extended to pairs of transfinite words in [4] where it is shown