Alternative loop

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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

Symbol-free definition

An alternative loop is an algebra loop that is also an alternative magma.

Definition with symbols

An alternative loop is an algebra loop $(L,*)$ satisfying the following two conditions:

• $\! x * (x * y) = (x * x) * y \ \forall \ x,y \in L$.
• $\! x * (y * y) = (x * y) * y \ \forall \ x,y \in L$.

Relation with other properties

Stronger properties

• Di-associative loop
• Moufang loop
• Group

Weaker structures

• Algebra loop
• Left-alternative loop
• Right-alternative loop
• Alternative magma
• Quasigroup