# 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
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