# Arithmetic of semigroups of series in multiplicative systems

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We study the arithmetic of a semigroup \(\mathcal{M}_{\mathcal{P}}\) of functions with operation of multiplication representable in the form \( f(x) = \sum\nolimits_{n = 0}^\infty {{a_n}{\chi_n}(x)\quad \left( {{a_n} \ge 0,\sum\nolimits_{n = 0}^\infty {{a_n} = 1} } \right)} \), where \( \left\{ {{\chi_n}} \right\}_{n = 0}^\infty \) is a system of multiplicative functions that are generalizations of the classical Walsh functions. For the semigroup \(\mathcal{M}_{\mathcal{P}}\), analogs of the well-known Khinchin theorems related to the arithmetic of a semigroup of probability measures in **R** ^{ n } are true. We describe the class \(I_0(\mathcal{M}_{\mathcal{P}})\) of functions without indivisible or nondegenerate idempotent divisors and construct a class of indecomposable functions that is dense in \(\mathcal{M}_{\mathcal{P}}\) in the topology of uniform convergence.

## Keywords

Probability Measure Multiplicative Function Compact Abelian Group Walsh Function Multiplicative System## References

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