Left inverse property implies two-sided inverses exist
Then, is the unique two-sided inverse of (in a weak sense) for all :
Note that it is not necessary that the loop be a right-inverse property loop, so it is not necessary that be a right inverse for in the strong sense.
- Equality of left and right inverses in monoid
- Two-sided inverse is unique if it exists in monoid
- Equivalence of definitions of inverse property loop
- Equivalence of definitions of gyrogroup
Given: A left-inverse property loop with left inverse map .
To prove: , where is the neutral element.
Proof: Putting in the left inverse property condition, we obtain that .
Next, putting , we obtain that:
Writing the on the right as and using cancellation, we obtain that:
This completes the proof.