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:

$(x * x) * y = x * (x * y) \forall x, y \in S$
$x * (y * y) = (x * y) * y \forall x, y \in S$

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	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