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We consider discrete time Markov chains in competition over a set of resources. We build a multidimensional Markov process based on the Cartesian product of the states space and on competition rules between the chains. When a resource is owned by a chain it affects the transition probabilities of the other components of the process. We prove that under some competition rules the steady-state distribution of the chain has a product form. This work extends Boucherie's theory based on continuous-time chains. The effects of the competition are slightly different from the restricted transitions studied by Boucherie. The proof is much more general and relies on algebraic properties of the generalized tensor product defined by Plateau and Stewart.