From Groupprops
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 is termed a magma in which cubes are welldefined if it satisfies the following equivalent conditions:
 For every , .
 Every element of commutes with its square.
The value is termed the cube of and is denoted by .
Relation with other properties
Stronger properties
Property 
Meaning 
Proof of implication 
Proof of strictness (reverse implication failure) 
Intermediate notions

Powerassociative magma 
all powers are welldefined 


Magma in which cubes and fourth powers are welldefined, Magma in which powers up to the fifth are welldefinedFULL LIST, MORE INFO

Commutative magma 
any two elements commute 


Flexible magma, Magma in which cubes are welldefined and every element commutes with its cubeFULL LIST, MORE INFO

Diassociative magma 
the submagma generated by any two elements is associative 


Alternative magma, Flexible magma, Left alternative magma, Powerassociative magma, Rightalternative magmaFULL LIST, MORE INFO

Semigroup 
associativity holds universally 


Alternative magma, Diassociative magma, Flexible magma, Left alternative magma, Powerassociative magma, Rightalternative magmaFULL LIST, MORE INFO

Jordan magma 
commutative, satisfies Jordan's identity 


Commutative magma, Magma in which cubes and fourth powers are welldefined, Magma in which powers up to the fifth are welldefinedFULL LIST, MORE INFO

Flexible magma 
satisfies the flexible law: 


Magma in which cubes are welldefined and every element commutes with its cubeFULL LIST, MORE INFO

Leftalternative magma 
satisfies the leftalternative law: 


FULL LIST, MORE INFO

Rightalternative magma 
satisfies the rightalternative law: 


FULL LIST, MORE INFO

Alternative magma 
both leftalternative and rightalternative 


Left alternative magma, Magma in which cubes and fourth powers are welldefined, Magma in which powers up to the fifth are welldefined, Rightalternative magmaFULL LIST, MORE INFO

Magma in which cubes and fourth powers are welldefined 



Magma in which cubes are welldefined and every element commutes with its cubeFULL LIST, MORE INFO

Magma in which powers up to the fifth are welldefined 



Magma in which cubes and fourth powers are welldefined, Magma in which cubes are welldefined and every element commutes with its cubeFULL LIST, MORE INFO

Magma in which cubes are welldefined and every element commutes with its cube 



FULL LIST, MORE INFO
