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


$$
A(T) = T.
$$



Arithmetic closed subsets
It does not matter much if one includes 1 or not. In the case that $T$ is finite and the members are square free you have the question: Given a family $T$ of subsets of a finite set $U$, what can we say about the smallest subset of the power-set $2^U$ which is closed under disjoint unions? I'm not sure how much there is to say. When $T$ is finite, also $A(T)$ is finite and a subset of the set of divisors of the least common multiple of the members of $T$. For $T$ infinite not so much changes. 


For fixed $k \gt 1,$ we have $ A(\lbrace t^k \mid t \in T \rbrace)=\lbrace a^k \mid a \in A(T) \rbrace$ but $A(kT)=kT.$ 


