Is there a name for this identity-like object? I was grading for a linear algebra class just now and someone remarked that since $U + U = U$, then $U$ would be an identity for the set of subspaces of some vector space with binary operation subspace addition. Obviously I corrected their mistake, since it isn't true that this fixed $U$ has the property $U + V = V$ for all subspaces $V$ of this vector space (except perhaps in trivial cases), but it made me curious whether or not there is a name for this type of object since they have exhibited a nontrivial instance of the existence of such an element.
I suppose it would only be meaningful in a semigroup or algebraic object with less structure than a semigroup, since the standard proof that the identity element is unique in, say, a group, would probably show that any element like this would have to be the identity. 
So, my question is, is there a name for an element of a semigroup (or an algebraic object with less structure if even a semigroup in general is too restrictive), let's call the element $x$ and the operation of the semigroup +, with the property that $x + x = x$.
 A: This property is called idempotence, and it can be nontrivial even in monoids; for example, there is a very large class of monoids all of whose elements are idempotent, namely any semilattice. In particular, the collection of subspaces of a vector space is an example. There are two corresponding idempotent monoid operations, namely intersection and sum.
Idempotents behave like identities in the following sense: if $m$ is an idempotent element of a semigroup $S$, you can consider the subsemigroup of elements $a$ satisfying $ma = am = a$. This is also a semigroup, even a monoid, and (by construction) has identity element $m$; in fact by construction it's the largest subsemigroup of $S$ on which $m$ acts as the identity.
When applied to an element $U$ of the lattice of subspaces of a vector space $V$, with monoid operation given by intersection, this construction produces the lattice of subspaces of $U$. With the monoid operation given by sum, this construction produces the lattice of subspaces of $V$ containing $U$, which can naturally be identified with the lattice of subspaces of $V/U$ by the isomorphism theorems. 
