• Source: Alternativity

In abstract algebra, alternativity is a property of a binary operation. A magma G is said to be left alternative if



(
x
x
)
y
=
x
(
x
y
)


{\displaystyle (xx)y=x(xy)}

for all



x
,
y
∈
G


{\displaystyle x,y\in G}

and right alternative if



y
(
x
x
)
=
(
y
x
)
x


{\displaystyle y(xx)=(yx)x}

for all



x
,
y
∈
G
.


{\displaystyle x,y\in G.}

A magma that is both left and right alternative is said to be alternative (flexible).
Any associative magma (that is, a semigroup) is alternative. More generally, a magma in which every pair of elements generates an associative submagma must be alternative. The converse, however, is not true, in contrast to the situation in alternative algebras. In fact, an alternative magma need not even be power-associative: already the expression



(
x
x
)
(
x
x
)


{\displaystyle (xx)(xx)}

cannot be proven to be identical to expressions such as



(
x
(
x
(
x
x
)
)
)


{\displaystyle (x(x(xx)))}

purely by alternativity.




See also


Flexible algebra
Power associativity




References

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