Is there a standard name for a set equipped only with an idempotent binary operation?

Question

Is there a name for an idempotent magma, or do they not arise often enough to warrant a special name?

(By idempotent binary operation, I mean an operation $+$ such that $x + x = x$ for any $x$.)

Answer 1

This doesn't seem to be a very interesting definition on its face; classification is pretty trivial: Given any set $S$, the idempotent binary operations $S \times S \to S$ are given by those functions which send $(x, x)$ to $x$ and behave arbitrarily on $(S \times S)-\Delta$, the nondiagonal elements.

Of course if you pile on a few additional axioms you might get something interesting, or maybe if you start examining relationships between different idempotent binary operations. I haven't heard of this kind of object being defined on its own though.