nLab power-associative algebra

Idea

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.

Definition

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.

Properties

The submagma of every power-associative invertible magma $M$ generated by an element $a \in M$ is a cyclic group. This means in particular there is a $\mathbb{Z}$-action on $M$ $(-)^{(-)}:M\times\mathbb{Z}\to M$ called the power.

Examples

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 Cayley–Dickson algebra, even beyond the octonions (the last alternative one), is power-associative.

See also

• polynomial function