Abstract
This article explores a generalization of the algebraic theory of formal languages. Having, as starting point, the work of T. Colcombet on cost functions and stabilization monoids, and of Daviaud et al. on stabilization algebras, this class of algebras is extended to ω♯-algebras and ω♯-automata are also introduced. The equality problem for order ideals (of free ω♯-algebras) recognized by finite ω♯-algebras is answered positively in this context. Various results on formal languages and monoids are generalized to this setting of order ideals and ω♯-algebras. The class of cost functions is proved to be embeddable in the class of recognizable order ideals.
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