Monoid of products of zeta functions of monoids of natural numbers
https://doi.org/10.22405/2226-8383-2022-23-3-102-117
Abstract
The paper studies algebraic structures arising with respect to the multiplication operationof two sets of natural numbers. The main objects of study are the monoid MN of monoids of natural numbers and the monoid SN of products of arbitrary subsets of a natural series. Also, the monoid will be SN* = SN ∖ {∅}.
An important property of these monoids is the fact that the set of all idempotents in the monoid SN except for the zero element coincides with the set of idempotents of the monoid SN* forms the monoid MN.
The presence of such a fact allowed us to consider the order. With respect to the order of 𝐴 6 𝐵 and binary operations inf, sup the monoid MN is an irregular, complete A-lattice.
The paper distinguishes the concepts of A-lattice as an object of general algebra and Tlattice as an object of number theory and geometry of numbers.The paper defines the structure of a complete metric space with a non-Archimedean metric on the monoid SN. This made it possible to prove a theorem on the convergence of a sequence of Dirichlet series over convergent sequences of natural numbers.
If we consider the product of two zeta functions of monoids of natural numbers, then it will be a zeta function of a monoid of natural numbers only when these monoids are mutually simple.
In general, their product will be a Dirichlet series with natural coefficients over a monoid equal to the product of the monoids of the cofactors. This monoid generated by the zeta functions of
the monoids of natural numbers is denoted by MD. It is shown that the monoids MN and MD are non-isomorphic.
The paper defines two small categories ℳ𝒩 and 𝒮𝒩 and studies some of their properties.
About the Authors
Mikhail Nikolaevich Dobrovol’skii
Geophysical centre of RAS
Russian Federation
Russian Federation
candidate of candidate of physical and mathematical sciences
Nikolai Nikolaevich Dobrovol’skii
Tula State Lev Tolstoy Pedagogical University, Tula State University
Russian Federation
Russian Federation
candidate of physical and mathematical sciences, associate professor
Nikolai Mihailovich Dobrovol’skii
Tula State Lev Tolstoy Pedagogical University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences, professor
Igor Borisovich Kozhukhov
NRU «Moscow Institute of Electronic Technology»
Russian Federation
Russian Federation
professor, doctor of physical and mathematical sciences
Irina Yuryevna Rebrova
Tula State Lev Tolstoy Pedagogical University
Russian Federation
Russian Federation
candidate of physical and mathematical sciences, associate professor
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Review
For citations:
Dobrovol’skii M.N., Dobrovol’skii N.N., Dobrovol’skii N.M., Kozhukhov I.B., Rebrova I.Yu. Monoid of products of zeta functions of monoids of natural numbers. Chebyshevskii Sbornik. 2022;23(3):102-117. (In Russ.) https://doi.org/10.22405/2226-8383-2022-23-3-102-117
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