Let $\Bbb{Z}_*$ be the monoid structure $x * y := Axy + x + y$ for some fixed $A \in\Bbb{N}$, $A \gt 1$.
Then $\Bbb{Z}_* \approx A\Bbb{Z} + 1$ is easy for me to prove on one line (see added Lemma 0 below).
I'm wondering if this means that $\Bbb{Z} \setminus \Bbb{Z}*\Bbb{Z} = \Bbb{Z}\setminus\{ Axy + x + y : x,y \in \Bbb{Z}\}$ is an infinite set.
By definition $\Bbb{Z}*\Bbb{Z}$ is closed under $*$, and contains $0$ so that $\Bbb{Z}*\Bbb{Z}$ is a submonoid of $\Bbb{Z}_*$.
I either want to take a congruence relation for a submonoid quotient in an abelian setting, or take the Grothendieck $G(\cdot)$ group of each.
Firstly, I don't know what congruence would be appropriate. And secondly, I don't know if $G(\Bbb{Z}_*) = \Bbb{Q}_*$ i.e. because I do know that $*$ gives $\Bbb{Q}$ a group structure isomorphic to $\Bbb{Q}^{\times}$ the group of units of the field $\Bbb{Q}$. And clearly this group has $\Bbb{Z}_*$ embedded into it as a submonoid.
Let me know if you'd like me to prove any of the above observations and I can extend this post some.
Lemma 0. $\psi(x) = Ax + 1$ defines an isomorphism of $\Bbb{Z}_*$ onto $A\Bbb{Z} + 1$ under $\times$.
Proof. $A(Axy + x + y) + 1 = (Ax + 1)(Ay + 1) \in A\Bbb{Z} + 1$. And clearly $\psi$ is onto and injective.
In any monoid $(M,*),$ with unit $e,$ the complement of $M*M\subseteq M$ is empty because $$M=M*e\subseteq M*M.$$