abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ' ...

, 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 ...

magma (that is, a

semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'' ...

) 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 algebra In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have *x(xy) = (xx)y *(yx)x = y(xx) for all ''x'' and ''y'' in the algebra. Every associative algebra is ...

s. In fact, an alternative magma need not even be power-associative.

References

Properties of binary operations {{algebra-stub