Spreading maps (polymorphisms), symmetries of Poisson processes and matching summation

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The matrix of a permutation is a partial case of Markov transition matrices. In the same way, a measure preserving bijection of a space A with finite measure is a partial case of Markov transition operators. A Markov transition operator also can be considered as a map (polymorphism) A to A, which spreads points of A into measures on A. In this paper, we discuss R-polymorphisms and -polymorphisms, who are analogues of the Markov transition operators for the groups of bijections A to A leaving the measure quasiinvariant; two types of the polymorphisms correspond to the cases, when A has finite and infinite measure respectively. We construct a functor from -polymorphisms to R-polymorphisms, it is described in terms of summation of convolution products of measures over matchings of Poisson configurations.