Power associativity

A binary operation $\star$ on a set $X$ is power associative if it satisfies the condition $$ x \star ( x \star x) = (x \star x) \star x $$ for all $x \in X$.

For such operations, exponentiation $x \mapsto x^n$ is well-defined for positive integers $n$, and $x^{m+n} = x^m \star x^n$. The set of powers of $x$ forms a semi-group.

See also: Algebra with associative powers.

References

• Bruck, Richard Hubert A survey of binary systems, Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge. 20 Springer (1958) Zbl 0081.01704