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Reduced Factorizations in Free Profinite Groups and Join Decompositions of Pseudovarieties

In this paper we investigate certain notions from the theory of free profinite groups, which we then apply to the study of certain pseudovarieties of monoids. More precisely, the profinite Cayley graphs of the free pro-H groups are H-trees for certain, so-called arborescent, pseudovarieties of groups H. For such pseudovarieties, we introduce the notion of reduced factorizations in the free pro-H groups, and we prove that these factorizations enjoy a common refinement property. This property, in turn, is used to compute joins of certain pseudovarieties of monoids with pseudovarieties of groups within the pseudovariety DS of all finite monoids in which each regular -class is a subsemigroup. We then apply our results to the theory of formal languages.