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The main motivation for investigating such entities comes from their connection with the classification of regular languages via Eilenberg\u2019s correspondence. This connection prompted the study of various natural operators on pseudovarieties and led to several important questions, both algebraic and algorithmic. The most important of these questions is decidability: given a finite semigroup is there an algorithm that tests whether it belongs to the pseudovariety? Since the most relevant operators on pseudovarieties do not preserve decidability, one often seeks to establish stronger properties. A key role is played by relatively free profinite semigroups, which is the counterpart of free algebras in universal algebra. 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