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I read from the linkhere proving that the inverse element in associative structure should be unique. However, it's only proving that should there exist a left and a right inverse element, then they should be identical. And I remember reading examples in non-commutative rings (so there should also be a similar case in monoids), an element has multiple multiplicative left inverse elements. But I'm not quite sure anymore. Could anybody reproduce such an example?
Also, interestingly, even for non-commutative groups, the inverse element is unique. Does all of this have to do with the fact that every element in the group is assigned with an inverse element?