---
tag: VEFE5WZMS3
kind: concept
name: Self-distributive operation
msc: [20N02, 08A62]
sources:
  - url: https://en.wikipedia.org/wiki/Distributive_property
---

## Summary

A self-distributive operation is a [[binary operation|0M418FE7BB]] that distributes over itself, on the left or on the right.
Left self-distributivity reads $x * (y * z) = (x * y) * (x * z)$.
Conjugation in a group, the join and meet of a lattice, and the operation $2x - y$ on an abelian group are self-distributive.
Left self-distributive magmas are called [[shelves|PDFC0MRYR8]]; together with racks and quandles they are used in knot theory and in set theory through the [[Laver tables|9H65YDXMGJ]].

## Explanations

### L1 human

The usual [[distributive law|0KBEMAJAYA]] says that multiplication spreads over addition: $a(b + c) = ab + ac$.
A self-distributive operation spreads over itself.
Acting with $x$ on a combination of $y$ and $z$ is the same as combining the results of acting with $x$ on each of them.
Think of $x * y$ as "move $y$ by the symmetry attached to $x$".
Then the rule says that moving a combined object is the same as moving its parts and combining them.
For example, $x * y = 2x - y$ reflects $y$ through $x$, and reflections behave in this way.