Monoid, where $2+2 = 2$

Let’s consider the set $S subset Bbb N$, consisting of all prime numbers. We can introduce a binary "addition" ($+$) operation on the power set $Bbb P(S)$, which is defined as the regular set union – so the set $Bbb P(S)$ becomes a monoid. It’s easy to see that all the elements of the set $Bbb P(S)$ can be thought as natural numbers, for which all the primes in their factorization aren’t repeated. So, adding $2$ and $2$ in this monoid gives us $2$:

$$2+2 = 2$$

Also, the number $4$ doesn’t belong to the $Bbb P(S)$, because its factorization contains $2$ to the second power.

Does this monoid have a name?