Left Bol loop

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This article defines a property that can be evaluated for a loop.
View other properties of loops

Definition

Definition with symbols

An algebra loop L with binary operation * is said to be a left Bol loop if it satisfies the following identity for all x, y, z \in L:

\! x * (y * (x * z)) = (x * (y * x)) * z

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Left Bruck loop |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Power-associative loop powers are well-defined |FULL LIST, MORE INFO
Left-inverse property loop left inverses (in a strong sense) are well-defined |FULL LIST, MORE INFO
Left alternative loop |FULL LIST, MORE INFO

Property obtained by the opposite operation

The dual notion to that of left Bol loop is that of a right Bol loop. The theory runs exactly the same.

Facts

Inverses

A left Bol loop has the property that the subloop generated by any element is a subgroup. Thus, we can define the inverse of an element in the left bol loop as its inverse in that subgroup. Also, it is clear that this is both the left and right inverse in the whole algebra loop.

Further, we can talk of the order of an element in a left Bol loop as the order of the subgroup generated by it.

References

  • Infinite simple Bol loops by Hubert Kiechle and Michael K. Kinyon
  • Finite Bruck loops by MIchael Aschbacher, Michael K. Kinyon, and J. D. Phillips

External links