Alternativity

In abstract algebra, alternativity is a property of a binary operation. A magma G is said to be left alternative if ${\displaystyle (xx)y=x(xy)}$ for all ${\displaystyle x,y\in G}$ and right alternative if ${\displaystyle y(xx)=(yx)x}$ for all ${\displaystyle x,y\in G}$. A magma that is both left and right alternative is said to be alternative.

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.