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 $x + x = x$.

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    $\begingroup$ Looks kind of like idempotence $\endgroup$ – G Tony Jacobs Sep 16 '17 at 13:42
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    $\begingroup$ When thinking about linear operators, $P^2 = P$ means $P$ is a projection. $\endgroup$ – Ethan Bolker Sep 16 '17 at 13:42
  • $\begingroup$ Ah! I was not at all thinking in terms of functions. I would think idempotence would be it! $\endgroup$ – Christian Sep 16 '17 at 13:47
  • $\begingroup$ You might be interested in the semiring of ideals associated to any ring. The set of ideals becomes a semiring using normal ideal addition and multiplication. It has idempotent addition. The multiplication becomes idempotent too if you use a von Neumann regular ring to start with. $\endgroup$ – rschwieb Sep 18 '17 at 11:08

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.


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