flexible algebraFlexibleAlgebra2004-10-10 14:34:052008-10-07 19:46:25Definitionleft powerright powerflexible% this is the default PlanetMath preamble. as your knowledge
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% define commands hereA non-associative algebra $A$ is \emph{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 \emph{left power} of $a$ iteratively as follows:
\begin{enumerate}
\item $L^1(a)=a$,
\item $L^n(a)=a\cdot L^{n-1}(a)$.
\end{enumerate}
Similarly, we can define the \emph{right power} of $a$ as:
\begin{enumerate}
\item $R^1(a)=a$,
\item $R^n(a)=R^{n-1}(a)\cdot a$.
\end{enumerate}
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.