Left Bol magma with neutral element

From Groupprops
Jump to: navigation, search
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 left Bol magma with neutral element is a left Bol magma that has a neutral element. In other words, it is a magma (S,*) such that:

  1. For all x,y,z \in S, we have \! x * (y * (x * z)) = (x * (y * x)) * z.
  2. There exists e \in S, we have \! x * e = e * x = x.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Left Bol loop
Moufang loop

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Flexible magma left Bol implies flexible
Left alternative magma left Bol implies left alternative