Monoid of products of zeta functions of monoids of natural numbers

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.