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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?

MattAllegro
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Malper
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  • Do you want it to self-distribute on both sides? Might you be assuming commutativity as well? – rschwieb Sep 12 '13 at 13:54
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    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
  • Great, thanks for the clarifying comment and edit! – rschwieb Sep 12 '13 at 13:56
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    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
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    @MartinBrandenburg Good observation. I recognize that condition: it's the flexible identity. – rschwieb Sep 12 '13 at 14:49
  • @Malper Can you comment on how close your proposed conditions bring you to being an idempotent rack? – rschwieb Sep 12 '13 at 14:52
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    @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
  • @Malper Interesting :) Given how many computational problems have already been solved with associative algebra, it's interesting to ponder what the next thousand years are going to bring with the study of other objects. That is, if we aren't vaporized during that time... – rschwieb Sep 12 '13 at 15:00
  • @ Martin Brandenburg: A ring is called alternative if every its 2-generated ring is associative. So here we have a special case of "alternative magmas". – Boris Novikov Oct 18 '13 at 20:21
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    These have also appeared in the literature under the name "spindle", regarded as generalisations of quandles. – James Dec 30 '13 at 06:29
  • Another observation @MartinBrandenburg: if we assume there's a special element $1$ floating around with $a \cdot 1 = 1$ and $1 \cdot a = a$, then the flexible identity implies idempotency, by taking $b$ to equal $1$. – goblin GONE Jan 28 '16 at 14:41
  • one might also call this an "idempotent shelf", as a magma with the self-distributivity property is known as a shelf. – May Emerson Mar 01 '25 at 20:45

2 Answers2

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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.

Boris Novikov
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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.

zyx
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  • Thank you for the reference! I look forward to reading about this topic. – Malper Oct 22 '13 at 13:32
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    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