# Monoid, where $2+2 = 2$

Mathematics Asked by HEKTO on October 3, 2020

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?

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