Geodesic
Dimension polynomials in the generalized difference case
Fundamentalʹnaâ i prikladnaâ matematika, Tome 17 (2012) no. 1, pp. 205-222.

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The aim of this paper is to generalize results about dimension polynomials of difference modules over difference rings for the wider class of rings of difference operators. We introduce the notion of the quasi-commutativity, which generalizes the notion of the commutativity and enables one to consider the wider classes of monoids and groups of endomorphisms. Some properties of quasi-commutative monoids and groups are established; these properties allow us to apply some methods that are almost similar to the ones used in working with free commutative monoids and groups. Also we prove the theorem of existence of the dimension polynomial of generalized difference modules in the cases where the submonoid of endomorphisms is free quasi-commutative. Also the existence of its analog for the case of a direct product of a free quasi-commutative monoid and a finite cyclic group is established. 
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S. N. Smirnov. Dimension polynomials in the generalized difference case. Fundamentalʹnaâ i prikladnaâ matematika, Tome 17 (2012) no. 1, pp. 205-222. http://geodesic.mathdoc.fr/item/FPM_2012_17_1_a11/

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