Semigroups defined by the subsets of groups In a group $G$, the non empty subsets form a semigroup (with identity) under the usual multiplication $ST=\{st \mid s\in S, t \in T\}$. 
This semigroup seems to be very rich of information, for instance if $G$ is finite the idempotents in this semigroup are exactly the subgroups of $G$.  Also for any (non empty) subset $S$, there is a power $S^n$ of $S$ which is an idempotent, and so a subgroup of $G$. if $S$ contains the identity, this would be the subgroup generated by $S$ (I don't know what happen if $S$ does not contain the identity).
1) Is there any attempts to studying such a semigroup, and its relations to the underlying group?
2) If $G$ is finite, our subsets can be identified to the elements of the group algebra $\mathbb{Z}_2[G]$, the multiplication in this algebra is somewhat different from the above one; can one covers the information about the semigroup of subsets of $G$ by only studying the group algebra $\mathbb{Z}_2[G]$?.
Thanks in advance. 
 A: The first question has been answered by a link in the comments. I will post it in an answer: What can we learn about a group by studying its monoid of subsets?
Indeed, as the second answer to that question says, these semigroups have been studied. They haven't been studied a lot because of the difficulty -- Lyapin's early book on semigroups has a paragraph on them in which the author dismisses the construction as too difficult to study. 
I find it interesting that you thought about the group algebra since I thought about it too. The question the link directs to is actually mine (I'm still getting a newsletter despite having deleted my account), and you can see the thought process that lead me to thinking about it there. The power semigroup, as it is called, is isomorphic to a certain variation on the group algebra idea, with $\mathbb Z_2$ replaced by $(\{0,1\},\vee,\wedge)$ and the requirement of finite supports dropped. With finite supports you get the so called finitary power semigroup (the semigroup if finite subsets), which has also been studied. This gives the idea of thinking of the elements of $\mathbb Z_2[G]$ as subsets of $G$ with a certain funny ring (or algebra) structure. I did find it fascinating when I noticed it and asked a certain professor about it, but my excitement was quelled by the professor's indifference. Power semigroups are pretty much the only mathematical structures I work on at the moment (it started with the linked question), but I have never seen any mention of even the similarity of the power semigroup construction to the group algebra construction in the literature.
For the question on orders in the comment, you might find this question interesting as it deals with exactly that in $\mathbb N$. 
