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.
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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

Weaker structures