In abstract algebra, an algebra is called alternative if (xx)y=x(xy) and y(xx)=(yx)x for all x and y in the algebra, that is, if the multiplication is alternative.
Equivalently, an algebra is alternative if and only if the subalgebra generated by any two of its elements is associative. The equivalence of the two definitions is known as Artin's Theorem.
For any two elements x and y in an alternative algebra another simple identity holds: (xy)x = x(yx). This is called the flexible law.
Every associative algebra is obviously alternative, but so too are some non-associative algebras such as the octonions.
Alternativity in algebras is a condition inbetween associativity and power associativity.
The algebras produced by this process are known as Cayley-Dickson algebras; since they extend the complex numbers, they are hypercomplex numbers.
These algebras all have a notion of norm and conjugate, with the general idea being that the product of an element and its conjugate should equal the square of its norm.
This algebra was discovered by Graves in 1844, and is called the octonions or the "Cayley numbers".
In abstract algebra, an algebra (or more generally a magma) is called alternative if the subalgebra generated by any two of its elements is associative.
An equivalent definition is to require, for all x and y in an algebra A, that x(xy) = (xx)y and (xy)y = x(yy).
Every associativealgebra is obviously alternative, but so too are some non-associative algebras such as the octonions.
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