The inverse problem for a basic monoid of type $q$

In the paper for an arbitrary basic monoid ${M(\mathbb{P}(q))}$ of type $q$ the inverse problem is solved, that is, finding the asymptotics for the distribution function of the elements of the monoid ${M(\mathbb{P}(q))}$, based on the asymptotics of the distribution of pseudo-prime numbers $\mathbb{P}(q)$ of type $q$.
To solve this problem, we consider two homomorphisms of the main monoid ${M(\mathbb{P}(q))}$ of type $q$ and the distribution problem reduces to the additive Ingham problem.
It is shown that the concept of power density does not work for this class of monoids. A new concept of $C$ logarithmic $\theta$-power density is introduced.
It is shown that any monoid ${M(\mathbb{P}(q))}$ for a sequence of pseudo-simple numbers $\mathbb{P}(q)$ of type $q$ has upper and lower bounds for the element distribution function of the main monoid ${M(\mathbb{P}(q))}$ of type $q$.
It is shown that if $C$ is a logarithmic $\theta$-power density for the main monoid ${M(\mathbb{P}(q))}$ of the type $q$ exists, then $\theta=\frac{1}{2}$ and for the constant $C$ the inequalities are valid $ \pi\sqrt{\frac{1}{3\ln q}}\le C\le \pi\sqrt{\frac{2}{3\ln q}}. $
The results obtained are similar to those previously obtained by the authors when solving the inverse problem for monoids generated by an arbitrary exponential sequence of primes of type $q$.
For basic monoids ${M(\mathbb{P}(q))}$ of the type $q$, the question remains open about the existence of a $C$ logarithmic $\frac{1}{2}$-power density and the value of the constant $C$.