Monoid of products of zeta functions of monoids of natural numbers

N. N. Dobrovol'skii ; M. N. Dobrovol'skii ; N. M. Dobrovol'skii ; I. B. Kozhukhov ; I. Yu. Rebrova

Čebyševskij sbornik, Tome 23 (2022) no. 3, pp. 102-117

Résumé

The paper studies algebraic structures arising with respect to the multiplication operation of two sets of natural numbers. The main objects of study are the monoid $\mathbb{MN}$ of monoids of natural numbers and the monoid $\mathbb{SN}$ of products of arbitrary subsets of a natural series. Also, the monoid will be $\mathbb{SN}^*=\mathbb{SN}\setminus\ptyset\$. An important property of these monoids is the fact that the set of all idempotents in the monoid $\mathbb{SN}$ except for the zero element coincides with the set of idempotents of the monoid $\mathbb{SN}^*$ forms the monoid $\mathbb{MN}$. The presence of such a fact allowed us to consider the order. With respect to the order of $A\le B$ and binary operations $\inf$, $\sup$ the monoid $\mathbb{MN}$ is an irregular, complete A-lattice. The paper distinguishes the concepts of A-lattice as an object of general algebra and T-lattice 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 $\mathbb{SN}$. 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 $\mathbb{MD}$. It is shown that the monoids $\mathbb{MN}$ and $\mathbb{MD}$ are non-isomorphic. The paper defines two small categories $\mathcal{MN}$ and $\mathcal{SN}$ and studies some of their properties.

Détail

Keywords: a monoid of natural numbers, a lattice by a monoid of natural numbers, a metric space of subsets of a natural series, a zeta function of a monoid, a Dirichlet series, a small category of monoids of natural numbers.