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 Magma in which powers up to the fifth are well-defined|FULL LIST, MORE INFO
Power-associative magma all positive powers are well-defined Magma in which powers up to the fifth are well-defined|FULL LIST, MORE INFO
Jordan magma 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 cubes well-defined, fourth powers need not be 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 cubes well-defined, every element commutes with its cube |FULL LIST, MORE INFO