In an associative algebra, or more generally in a semigroup, there is no difficulty in defining for any element and any positive natural number (as well as when the algebra has an identity element). In a nonassociative algebra, or more generally a magma, we can have many definitions, starting with (which could be but could be instead).
However, it may be that all possible ways of parenthesizing the expression for are equal. In this case, we call the algebra power-associative.
Let be a multicategory, and let be a magma object in , that is an object equipped with a bimorphism (binary multimorphism) . Consider the submulticategory of generated by and ; for each natural number , it has a family of -ary multimorphisms whose size is the Catalan number of .
The magma object is power-associative if, for each , all of these -ary multimorphisms are equal. Traditionally, we make no requirement for , since , but arguably we should require a unique -ary multimorphism in the subalgebra too; then we get a power-associative magma with an identity element.
If is Set, then a magma object is simply a magma, and we have a power-associative magma. If is -Mod for a commutative ring, then a magma object is a nonassociative algebra over , and we have a power-associative algebra.
The submagma of every power-associative invertible magma generated by an element is a cyclic group. This means in particular there is a -action on called the power.
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 CayleyβDickson algebra, even beyond the octonions (the last alternative one), is power-associative.
Last revised on August 21, 2024 at 02:16:56. See the history of this page for a list of all contributions to it.