I was playing around with the following object: Let $Q$ be a set with a binary operator $\cdot$ obeying the axioms:
$a \cdot a = a$ (idempotence)
$a \cdot (b \cdot c) = (a \cdot b) \cdot (a \cdot c)$ (left self-distributivity)
Examples of this would be group conjugation, semilattices, and quandles in knot theory. Does this general algebraic object have a name, and has it been studied?