Let $S, S_1$ be subsets of the positive numbers $\mathbb{N}$. We say (as usual) that $S$ is multiplicative closed if $x \in S$ and $y \in S$ implies $xy \in S.$ We say also that $S_1$ is arithmetic closed if $x \in S_1$ and $y \in S_1,$ and $\gcd(x,y)=1,$ implies $xy \in S_1.$

Let $T$ be a subset of the positive integers containing $1$ and at least another element. The smallest multiplicative closed set that contains $T$ (say $C(T)$ ) i.e., the intersection of all multiplicative closed sets that contain $T,$ is also equal to the set containing $1$ and containing all finite products of powers of primes $p^{a_p}$ (say), where $p$ is prime and $a_p >0$ is minimal with the property $p^{a_p} \in T.$

Question: How to describe (analogously ?) the smallest arithmetic closed set that contains $T$ (say $A(T)$ ) i.e., the intersection of all arithmetic closed sets that contain $T.$

Example: If $T$ contains exactly $1$ and all $3 \cdot 2^n$ with $n>0$ then $C(T)$ is the set of $1$ and all products $2^a \cdot 3^b$ with $a >0$ and $b > 0,$ while

$$ A(T) = T. $$