| Summary: | We consider implicit signatures over finite semigroups determined by sets of
pseudonatural numbers. We prove that, under relatively simple hypotheses on a
pseudovariety V of semigroups, the finitely generated free algebra for the
largest such signature is closed under taking factors within the free pro-V
semigroup on the same set of generators. Furthermore, we show that the natural
analogue of the Pin-Reutenauer descriptive procedure for the closure of a
rational language in the free group with respect to the profinite topology
holds for the pseudovariety of all finite semigroups. As an application, we
establish that a pseudovariety enjoys this property if and only if it is full.
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