Alternative magma

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 an alternative magma if it is both a left-alternative magma and a right-alternative magma, i.e., it satisfies the following two identities:

• $(x*x)*y=x*(x*y)\mathrm{\forall }x,y\in S$
• $x*(y*y)=(x*y)*y\mathrm{\forall }x,y\in S$

Relation with other properties

Property obtained by the opposite operation

Suppose $(S,*)$ is a magma and we define $\cdot$ on $S$ as $a\cdot b:=b*a$. Then, $(S,*)$ is an alternative magma if and only if $(S,\cdot )$ is an alternative magma.

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Diassociative magma	submagma generated by any two elements is associative			|FULL LIST, MORE INFO
Semigroup	whole magma is associative			|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Left-alternative magma	$x*(x*y)=(x*x)*y$			|FULL LIST, MORE INFO
Right-alternative magma	$x*(y*y)=(x*y)*y$			|FULL LIST, MORE INFO
Magma in which cubes are well-defined	$x*(x*x)=(x*x)*x$			|FULL LIST, MORE INFO
Magma in which powers up to the fifth are well-defined	$x^3,x^4,x^5$ well-defined for all $x$	alternative implies powers up to the fifth are well-defined		|FULL LIST, MORE INFO