Magma in which powers up to the fifth are well-defined

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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 (S,*) is termed a magma in which powers up to the fifth are well-defined if it satisfies the following three conditions for all x \in S (here, we denote x * x by x^2):

  1. Cubes are well-defined, i.e., x^2 * x = x * x^2. The common value is denoted x^3.
  2. Fourth powers are well-defined, i.e., x^3 * x = x^2 * x^2 = x * x^3. The common value is denoted x^4.
  3. Fifth powers are well-defined, i.e., x^4 * x = x^3 * x^2 = x^2 * x^3 = x * x^4. The common value is denoted x^5.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Power-associative magma all positive powers are well-defined (obvious) |FULL LIST, MORE INFO
Alternative magma satisfies the left-alternative and right-alternative laws alternative implies powers up to the fifth are well-defined |FULL LIST, MORE INFO
Jordan magma commutative, satisfies Jordan's identity Jordan implies 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 and fourth powers are well-defined cubes, fourth powers well-defined (obvious) |FULL LIST, MORE INFO
Magma in which cubes are well-defined cubes are well-defined (obvious) 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