Geodesic
Indices of imprimitivity of the temporal components of a semigroup of nonnegative matrices
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXXI, Tome 472 (2018), pp. 17-30

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It is proved that the index of imprimitivity of a semigroup of nonnegative block-monominal matrices free of zero rows decomposes into a sum of the indices of imprimitivity of its temporal components, and if the semigroup is block irreducible, then the indices of imprimitivity of all the temporal components are equal. 
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     author = {Yu. A. Al'pin and V. S. Al'pina},
     title = {Indices of imprimitivity of the temporal components of a semigroup of nonnegative matrices},
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Yu. A. Al'pin; V. S. Al'pina. Indices of imprimitivity of the temporal components of a semigroup of nonnegative matrices. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXXI, Tome 472 (2018), pp. 17-30. http://geodesic.mathdoc.fr/item/ZNSL_2018_472_a1/