flexible algebra
A nonassociative algebra $A$ is flexible if $[a,b,a]=0$ for all $a,b\in A$, where $[,,]$ is the associator on $A$. In other words, we have $(ab)a=a(ba)$ for all $a,b\in A$. Any associative algebra is clearly flexible. Furthermore, any alternative algebra^{} with characteristic $\ne 2$ is flexible.
Given an element $a$ in a flexible algebra $A$, define the left power of $a$ iteratively as follows:

1.
${L}^{1}(a)=a$,

2.
${L}^{n}(a)=a\cdot {L}^{n1}(a)$.
Similarly, we can define the right power of $a$ as:

1.
${R}^{1}(a)=a$,

2.
${R}^{n}(a)={R}^{n1}(a)\cdot a$.
Then, we can show that ${L}^{n}(a)={R}^{n}(a)$ for all positive integers $n$. As a result, in a flexible algebra, one can define the (multiplicative) power of an element $a$ as ${a}^{n}$ unambiguously.
Title  flexible algebra 

Canonical name  FlexibleAlgebra 
Date of creation  20130322 14:43:30 
Last modified on  20130322 14:43:30 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  11 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 17A20 
Related topic  Associator 
Related topic  AlternativeAlgebra 
Defines  left power 
Defines  right power 
Defines  flexible 