Simply put, is there a groupoid whose element can have multiple inverse elements? I know how to prove that elements of a semigroup have unique inverses, but this is a bit different... If there is such a groupoid, can someone give an example?
2026-03-25 05:59:18.1774418358
Groupoid element with multiple inverse elements?
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No. An inverse of $a$ is an element $b$ such that $cab=c,abc'=c',dba=d,$ and $bad'=d'$ whenever those elements are defined. But then if $b'$ is another inverse, $b'ab=b'=b$. In face this shows more generally that in a "monoidoid," where every element need not have an inverse, having a left and a right inverse is equivalent to being invertible, so that the left and right inverses must coincide.