An Eilenberg Theorem for Words on Countable Ordinals

We present in this paper an algebraic approach to the theory of languages of words on countable ordinals. The algebraic structure used, called an Ω1-semigroup, is an adaptation of the one used in the theory of regular languages of Ω-words. We show that finite Ω1-semigroups are equivalent to automata. In particular, the proof gives a new algorithm for determinizing automata on countable ordinals. As in the cases of finite and Ω-words, a syntactic Ω1-semigroup can effectively be associated with any regular language of words on countable ordinals. This result is used to prove an Eilenberg type theorem. There is a one-to-one correspondence between varieties of Ω1-languages and pseudo-varieties of Ω1-semigroups.

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