Magma in which powers up to the fifth are well-defined

This article defines a property that can be evaluated for a magma, and is invariant under isomorphisms of magmas.
View other such properties

Definition

A magma $(S,*)$ is termed a magma in which powers up to the fifth are well-defined if it satisfies the following three conditions for all $x\in S$ (here, we denote $x*x$ by $x^{2}$):

1. Cubes are well-defined, i.e., $x^{2}*x=x*x^{2}$. The common value is denoted $x^{3}$.
2. Fourth powers are well-defined, i.e., $x^{3}*x=x^{2}*x^{2}=x*x^{3}$. The common value is denoted $x^{4}$.
3. Fifth powers are well-defined, i.e., $x^{4}*x=x^{3}*x^{2}=x^{2}*x^{3}=x*x^{4}$. The common value is denoted $x^{5}$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Power-associative magma	all positive powers are well-defined	(obvious)		|FULL LIST, MORE INFO
Alternative magma	satisfies the left-alternative and right-alternative laws	alternative implies powers up to the fifth are well-defined		|FULL LIST, MORE INFO
Jordan magma	commutative, satisfies Jordan's identity	Jordan implies powers up to the fifth are well-defined		|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Magma in which cubes and fourth powers are well-defined	cubes, fourth powers well-defined	(obvious)		|FULL LIST, MORE INFO
Magma in which cubes are well-defined	cubes are well-defined	(obvious)		|FULL LIST, MORE INFO