Two-sided inverse is unique if it exists in monoid

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Suppose S is a monoid with binary operation * and identity element (neutral element) e. Suppose a \in S has a two-sided inverse b, i.e., b * a = a * b = e. Then, b is the only two-sided inverse for a, i.e., if c is an element such that c * a = a * c = e, then b = c.

Facts used

  1. Equality of left and right inverses in monoid


Fact (1) says that every left inverse must equal every right inverse. Thus, if we have two two-sided inverses, we can treat one of them as a left inverse and the other as a right inverse, forcing both of them to be equal.