In abstract algebra, alternativity is a property of a

binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary op ...

. A magma ''G'' is said to be if

(xx)y = x(xy)

for all

x, y \in G

and if

y(xx) = (yx)x

for all

x, y \in G.

A magma that is both left and right alternative is said to be ().. Any

associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...

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.

References

Properties of binary operations {{algebra-stub