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

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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 Magma in which cubes and fourth powers 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