Magma in which cubes 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 are well-defined if it satisfies the following equivalent conditions:

• For every $x\in S$, $x*(x*x)=(x*x)*x$.
• Every element of $S$ commutes with its square.

The value $x*(x*x)$ is termed the cube of $x$ and is denoted by $x^{3}$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Power-associative magma	all powers are well-defined			|FULL LIST, MORE INFO
Commutative magma	any two elements commute			|FULL LIST, MORE INFO
Diassociative magma	the submagma generated by any two elements is associative			|FULL LIST, MORE INFO
Semigroup	associativity holds universally			|FULL LIST, MORE INFO
Jordan magma	commutative, satisfies Jordan's identity			|FULL LIST, MORE INFO
Flexible magma	satisfies the flexible law: $x*(y*x)=(x*y)*x$			|FULL LIST, MORE INFO
Left-alternative magma	satisfies the left-alternative law: $(x*x)*y=x*(x*y)$			|FULL LIST, MORE INFO
Right-alternative magma	satisfies the right-alternative law: $x*(y*y)=(x*y)*y$			|FULL LIST, MORE INFO
Alternative magma	both left-alternative and right-alternative			|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
Magma in which cubes are well-defined and every element commutes with its cube				|FULL LIST, MORE INFO