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Alternate Algebra

Let $A$ denote an $\Bbb{R}$-Algebra, so that $A$ is a Vector Space over $R$ and

A\times A\to A
\end{displaymath} (1)

(x,y)\mapsto x\cdot y.
\end{displaymath} (2)

Then $A$ is said to be alternate if, for all $x,y\in A,$
(x\cdot y)\cdot y=x\cdot(y\cdot y)
\end{displaymath} (3)

(x\cdot x)\cdot y=x\cdot(x\cdot y).
\end{displaymath} (4)

Here, Vector Multiplication $x\cdot y$ is assumed to be Bilinear.


Finch, S. ``Zero Structures in Real Algebras.''

Schafer, R. D. An Introduction to Non-Associative Algebras. New York: Dover, 1995.

© 1996-9 Eric W. Weisstein