Magma in which cubes are well-defined and every element commutes with its cube

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 in which cubes are well-defined and every element commutes with its cube is a magma $(S,*)$ satisfying the following two conditions:

1. For every $a\in S$, $a$ commutes with the value $a^{2}=a*a$. In other words, $a*a^{2}=a^{2}*a$. This common value is denoted $a^{3}$.
2. For every $a\in S$, $a*a^{3}=a^{3}*a$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Commutative magma	any two elements commute			|FULL LIST, MORE INFO
Flexible magma	$x*(y*x)=(x*y)*x$	flexible implies cubes are well-defined and every element commutes with its cube		|FULL LIST, MORE INFO
Magma in which cubes and fourth powers are well-defined				|FULL LIST, MORE INFO
Magma in which 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 are well-defined				|FULL LIST, MORE INFO