What is this algebraic object called?

Question

I was playing around with the following object: Let $Q$ be a set with a binary operator $\cdot$ obeying the axioms:

1. $a \cdot a = a$ (idempotence)

2. $a \cdot (b \cdot c) = (a \cdot b) \cdot (a \cdot c)$ (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?

Answer 1

There have been a lot of papers on this subject after Patrick Dehornoy connected it to extensions and orderings of braid groups. His book Braids and Self-Distributivity is a canonical and very well written reference.

Dehornoy uses the terms LD- and LDI-systems. People who had studied the combinatorics of the same axioms (with a second operation) that arise in "algebras" of elementary embeddings in set theory, called them LD and LDI algebras.

Where LD=left (self) distributive and I=idempotent.

Answer 2

V.D. Belousov [Foundations of the theory of quasi-groups and loops , Moscow (1967) (In Russian)] called quasigroups with the axiom $2$ left distributive. So you can call your object an idempotent left distributive groupoid/magma.