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 Magma in which cubes and fourth powers are well-defined, Magma in which powers up to the fifth are well-defined|FULL LIST, MORE INFO
Commutative magma any two elements commute Flexible magma, Magma in which cubes are well-defined and every element commutes with its cube|FULL LIST, MORE INFO
Diassociative magma the submagma generated by any two elements is associative Alternative magma, Flexible magma, Left alternative magma, Power-associative magma, Right-alternative magma|FULL LIST, MORE INFO
Semigroup associativity holds universally Alternative magma, Diassociative magma, Flexible magma, Left alternative magma, Power-associative magma, Right-alternative magma|FULL LIST, MORE INFO
Jordan magma commutative, satisfies Jordan's identity Commutative magma, Magma in which cubes and fourth powers are well-defined, Magma in which powers up to the fifth are well-defined|FULL LIST, MORE INFO
Flexible magma satisfies the flexible law: $x * (y * x) = (x * y) * x$ Magma in which cubes are well-defined and every element commutes with its cube|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 Left alternative magma, Magma in which cubes and fourth powers are well-defined, Magma in which powers up to the fifth are well-defined, Right-alternative magma|FULL LIST, MORE INFO
Magma in which cubes and fourth powers are well-defined Magma in which cubes are well-defined and every element commutes with its cube|FULL LIST, MORE INFO
Magma in which powers up to the fifth are well-defined Magma in which cubes and fourth powers are well-defined, Magma in which cubes are well-defined and every element commutes with its cube|FULL LIST, MORE INFO
Magma in which cubes are well-defined and every element commutes with its cube |FULL LIST, MORE INFO