# What is this algebraic object called?

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?

• Neither. Note that group conjugation is only left self-distributive. Also, if you make this object commutative then it becomes a semilattice. – Malper Sep 12 '13 at 13:55
• Just one observation: 1. and 2. imply $a(ba)=(ab)a$. This is a weak form of associativity. Probably this property already has a name? – Martin Brandenburg Sep 12 '13 at 14:22
• @MartinBrandenburg Good observation. I recognize that condition: it's the flexible identity. – rschwieb Sep 12 '13 at 14:49
• @rschwieb An idempotent rack (a.k.a. a quandle) also has the property that the action of each element under left multiplication is a bijection. This object is more general. For example, if you had $a\cdot b = a$ for all $a, b \in Q$ ($\left|Q\right| > 1$), it would satisfy this definition but not be a quandle. – Malper Sep 12 '13 at 14:57
• These have also appeared in the literature under the name "spindle", regarded as generalisations of quandles. – James Dec 30 '13 at 6:29

## 2 Answers

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.

• Thank you for the reference! I look forward to reading about this topic. – Malper Oct 22 '13 at 13:32
• That field had a brilliant opportunity to bring terms like LSD formula, LSD identity, LSD systems, ... to mathematics. Dehornoy does write it out as left self distributivity. – zyx Oct 22 '13 at 15:39

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.