---
tag: PDFC0MRYR8
kind: concept
name: Shelf
msc: [20N02, 57K12]
specializes: [9689SFPVE9]
sources:
  - url: https://en.wikipedia.org/wiki/Racks_and_quandles
---

## Summary

A shelf is a set with a left [[self-distributive operation|VEFE5WZMS3]], that is, a [[magma|9689SFPVE9]] in which each left translation is an endomorphism.
A [[rack|X74VY86TSX]] is a shelf in which each left translation is a bijection.
A quandle is a rack in which $x \triangleright x = x$ for every element $x$.
Shelves that are not racks include the [[Laver tables|9H65YDXMGJ]], which come from set theory, and free shelves, whose study is linked to the braid groups.

## Explanations

### L1 human

In a shelf, every element $x$ acts on the other elements by $y \mapsto x \triangleright y$.
The only rule is that this action respects the operation itself: acting with $x$ on $y \triangleright z$ is the same as letting $x$ act on $y$ and on $z$ first.
Conjugation in a group is the model: conjugating a conjugate is the conjugate of the conjugates.
A simple example is any set with $x \triangleright y = y$, where every element acts as the identity.
Unlike in a rack, the action of an element in a shelf may collapse different elements together.