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## Main Question or Discussion Point

Hello

I have a question about the uniqueness of the inverse element in a groupoid. When we were in class our profesor wrote ##\text{Let} (M,*) \,\text{be a monoid then the inverse (if it exists) is unique}##. He then went off to prove that and I understood it, however I got curious and started thinking if it is possible to prove that there is only one unique inverse without taking into account the associative property of the semigroup. So then I started trying to prove it but I didn't really get too far and I tried looking online and also didn't find much about it. Could anybody tell me how this would be done ?

thanks

I have a question about the uniqueness of the inverse element in a groupoid. When we were in class our profesor wrote ##\text{Let} (M,*) \,\text{be a monoid then the inverse (if it exists) is unique}##. He then went off to prove that and I understood it, however I got curious and started thinking if it is possible to prove that there is only one unique inverse without taking into account the associative property of the semigroup. So then I started trying to prove it but I didn't really get too far and I tried looking online and also didn't find much about it. Could anybody tell me how this would be done ?

thanks

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