Diassociative 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 a diassociative magma if it satisfies the following equivalent conditions:

1. For any (possibly equal) elements $x,y\in S$, the submagma of $S$ generated by $x$ and $y$ is associative.
2. Any subset of the magma of size at most two is contained in a subsemigroup of the magma.
3. For any word in two letters, and any two different ways of associating (parethensizing) that word, the value of the result of the two ways of parenthesizing is the same for any choice of values of elements in the magma to substitute for the letters. In other words, all identities that would follow from the associative law hold in the magma as long as there are only two distinct letters in the statement of the identity.

Relation with other properties

Property obtained by the opposite operation

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

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Semigroup	associativity holds on the whole magma			|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
Alternative magma				|FULL LIST, MORE INFO
Power-associative magma	powers are well-defined			|FULL LIST, MORE INFO
Flexible magma	$x*(y*x)=(x*y)*x$			|FULL LIST, MORE INFO
Magma in which cubes are well-defined	$x*(x*x)=(x*x)*x$			|FULL LIST, MORE INFO