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Alternative identity

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A condition on a binary operation $\cdot$ on a set $X$: that for all $x, y \in X$ $$ x \cdot (x \cdot y) = (x \cdot x) \cdot y $$ and $$ x \cdot (y \cdot y) = (x \cdot y) \cdot y \ . $$

Alternative rings and algebras are those whose multiplication satisfies the alternative identity, which in this case is equivalent to the associator $(x,y,z)$ being an alternating function of three variables.

How to Cite This Entry:
Alternative identity. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Alternative_identity&oldid=37377