Left-invertible elements of monoid form submonoid
Let be a monoid with neutral element . Then, the set of left-invertible elements of form a submonoid of .
Inverse map is involutive: Specifically the reversal law for inverses.
Given: A monoid with neutral element
To prove: is left-invertible, and further, if are left-invertible elements of , so is
Proof: Clearly, is left-invertible, since .
Suppose are left-invertible and are left inverses for . Then consider:
Thus, is a left inverse for , so is left-invertible, completing the proof.