Rack

This article defines a property that can be evaluated for a magma, and is invariant under isomorphisms of magmas.
View other such properties

Definition

Note that the notion is not left-right symmetric. The definition given here implicitly is the version more suited to right actions.

In infix notation

A rack is a set $S$ with a binary operation $*$ satisfying the following

1. For every $a,b\in S$, there is a unique $c$ such that $a=c*b$.
2. The rack identity: For all $a,b,c\in S$, we have $\!(a*b)*c=(a*c)*(b*c)$. This identity may also be called right autodistributivity indicating that the operation right-distributes over itself.

In exponential notation

Here, we denote $a*b$ by $a^{b}$, and $a^{bc}$ stands for $(a*b)*c$. Then, the two conditions are:

1. For every $a,b\in S$, there is a unique $c$ such that $a=c^{b}$.
2. The rack identity:For all $a,b,c\in S$, we have $\!a^{bc}=a^{cb^{c}}$.

More abstract sense

A rack is a magma in which, for every element, the right multiplication by that element defines an automorphism of the magma. (Condition (1) guarantees bijectivity, and condition (2) is the homomorphism condition).

Related notions

• Quandle is a rack in which every element is idempotent, i.e., $a*a=a$ for all $a\in S$.
• Conjugation rack of a group: For any group $G$, we can turn $G$ into a rack by defining $a^{b}=b^{-1}ab$, i.e., the right action by conjugation.