If [itex]M[/itex] is a monoid and [itex]u,v\in M[/itex], and [itex]uv = 1[/itex], do we know [itex]vu = 1[/itex]? Can someone prove this or provide a counterexample? I tried to come up with one (a counterexample, that is) using 2 x 2 matrices but was unsuccessful.
Assume: [tex] uv=1[/tex] If there exists a w such that: [tex]vw=1[/tex] Then by associativity: [tex]w=1w=(uv)w=u(vw)=u1=u[/tex] Thus if such a w exists it must be u. This isn't quite enough but I can't find a short proof either. I'll think on it.
The canonical example is left and right shift of a sequence or countable dimensional vector space. L(a,b,c,d,...) = (b,c,d,....) R(a,b,c,..)=(0,a,b,c...) LR=id, and RL=/=id. You won't find one in 2x2 matrices - the invertible ones form a group, so there's no point looking.