Geodesic
Factoriality and the Pin-Reutenauer procedure
Discrete mathematics & theoretical computer science, Tome 18 (2015-2016) no. 3.

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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. 
@article{DMTCS_2016_18_3_a2,
     author = {Almeida, J. and Costa, J. C. and Zeitoun, M.},
     title = {Factoriality and the {Pin-Reutenauer} procedure},
     journal = {Discrete mathematics & theoretical computer science},
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     volume = {18},
     number = {3},
     year = {2015-2016},
     doi = {10.46298/dmtcs.650},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.650/}
}
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Almeida, J.; Costa, J. C.; Zeitoun, M. Factoriality and the Pin-Reutenauer procedure. Discrete mathematics & theoretical computer science, Tome 18 (2015-2016) no. 3. doi : 10.46298/dmtcs.650. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.650/

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