Magma in which cubes and fourth powers 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 cubes and fourth powers are well-defined if it satisfies the following conditions for all $a\in S$ (here, $a^{2}$ is shorthand for $a*a$):

1. Cubes are well-defined: $a^{2}*a=a*a^{2}$, and the common value is denoted $a^{3}$.
2. Fourth powers are well-defined: $a^{3}*a=a^{2}*a^{2}=a*a^{3}$, and the common value is denoted $a^{4}$.

Relation with other properties

Stronger properties

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

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Magma in which cubes are well-defined	cubes well-defined, fourth powers need not be			|FULL LIST, MORE INFO
Magma in which cubes are well-defined and every element commutes with its cube	cubes well-defined, every element commutes with its cube			|FULL LIST, MORE INFO