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In an associative algebra, or more generally in a semigroup, there is no difficulty in defining $x^n$ for $x$ any element and $n$ any positive natural number (as well as $x^0$ when the algebra has an identity element). In a nonassociative algebra, or more generally a magma, we can have many definitions, starting with $x^3$ (which could be $(x x) x$ but could be $x (x x)$ instead).
However, it may be that all possible ways of parenthesizing the expression for $x^n$ are equal. In this case, we call the algebra power-associative.
Let $\mathcal{C}$ be a multicategory, and let $A$ be a magma object in $\mathcal{C}$, that is an object $A$ equipped with a bimorphism (binary multimorphism) $m\colon A, A \to A$. Consider the submulticategory of $\mathcal{C}$ generated by $A$ and $m$; for each natural number $n$, it has a family of $n$-ary multimorphisms whose size is the Catalan number $C(n)$ of $n$.
The magma object $A$ is power-associative if, for each $n$, all of these $n$-ary multimorphisms are equal. Traditionally, we make no requirement for $n = 0$, since $C(0) = 0$, but arguably we should require a unique $0$-ary multimorphism in the subalgebra too; then we get a power-associative magma with an identity element.
If $\mathcal{C}$ is Set, then a magma object is simply a magma, and we have a power-associative magma. If $\mathcal{C}$ is $K$-Mod for $K$ a commutative ring, then a magma object is a nonassociative algebra over $K$, and we have a power-associative algebra.
Every associative algebra or semigroup is of course power-associative. More generally, every alternative algebra is also power-associative. Every Jordan algebra, although not necessarily alternative, is power-associative. Every CayleyโDickson algebra, even beyond the octonions (the last alternative one), is power-associative.
Last revised on September 5, 2016 at 06:09:25. See the history of this page for a list of all contributions to it.