Schützenberger and Eilenberg theorems for words on linear orderings

International audienceThis paper contains extensions to words on countable scattered linear orderings of two well-known results of characterization of languages of finite words. We first extend a theorem of Schützenberger establishing that the star-free sets of finite words are exactly the languages recognized by finite aperiodic semigroups. This gives an algebraic characterization of star-free sets of words over countable scattered linear orderings. Contrarily to the case of finite words, first-order definable languages are strictly included into the star-free languages when countable scattered linear orderings are considered. Second, we extend the variety theorem of Eilenberg for finite words: there is a one-to-one correspondence between varieties of languages of words on countable scattered linear orderings and pseudo-varieties of algebras. The star-free sets are an example of such a variety of languages

Eilenberg Theorems for Free

Eilenberg-type correspondences, relating varieties of languages (e.g. of finite words, infinite words, or trees) to pseudovarieties of finite algebras, form the backbone of algebraic language theory. Numerous such correspondences are known in the literature. We demonstrate that they all arise from the same recipe: one models languages and the algebras recognizing them by monads on an algebraic category, and applies a Stone-type duality. Our main contribution is a variety theorem that covers e.g. Wilke's and Pin's work on $\infty$-languages, the variety theorem for cost functions of Daviaud, Kuperberg, and Pin, and unifies the two previous categorical approaches of Boja\'nczyk and of Ad\'amek et al. In addition we derive a number of new results, including an extension of the local variety theorem of Gehrke, Grigorieff, and Pin from finite to infinite words

Schützenberger and Eilenberg theorems for words on linear orderings

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