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# flexible algebra

A non-associative 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 $\neq 2$ 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. $L^{n}(a)=a\cdot L^{{n-1}}(a)$.

Similarly, we can define the *right power* of $a$ as:

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

2. $R^{n}(a)=R^{{n-1}}(a)\cdot a$.

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.

## Mathematics Subject Classification

17A20*no label found*

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