Can a monoid have at least 2 members for which $xy = 1$ but $yx \neq 1$ ?
I tried matrix multiplication but $ AX = I$ then $XA = I $ too.
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Related questions: In a non-commutative monoid, is the left inverse of an element also the right inverse?, Left Invertible Elements of a monoid. – J.-E. Pin Feb 21 '16 at 18:00
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Yes, the bicyclic monoid, which is the monoid on two generators $a$ and $b$ under the relation $ab = 1$. Then $ba \not=1$. See this answer for more details.
