For a monoid, if uv = 1, do we know vu = 1?

  1. 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.
     
  2. jcsd
  3. jambaugh

    jambaugh 1,802
    Science Advisor
    Gold Member

    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.
     
  4. matt grime

    matt grime 9,395
    Science Advisor
    Homework Helper

    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.
     
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