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Math Help - a generalization of unit elements

  1. #1
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    a generalization of unit elements

    If I have a monoid S, and an element u\in S, then the following equivalence holds:

    u is a unit iff f(x)=xu and g(x)=ux are bijections onto S.

    Proof. Suppose u is a unit. Let x,y\in S. Then we have

    1. x=xu^{-1}u=f(xu^{-1});
    2. if f(x)=f(y), then xu=yu so x=xuu^{-1}=yuu^{-1}=y.

    Conversely, if f and g are bijections onto S, then there exists x\in S such that xu=f(x)=1 and y\in S such that ux=g(x)=1. Thus u has a left inverse and a right inverse, which is known to imply that u is a unit.

    Now let's say S is a semigroup without an identity element. Can there be an element u\in S such that f(x)=xu and g(x)=ux are bijections? Is it possible when S is finite? (Then it's equivalent to asking whether a semigroup without ideantity can have a cancellable element.) Is it possible when S is infinite? Is it possible when we only demand that at least one of f,g be a bijection onto S?
    Last edited by ymar; September 12th 2012 at 03:11 AM.
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  2. #2
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    Re: a generalization of unit elements

    a partial answer:

    if such bijections x→xu and x→ux exists for some element u, then for some element x:

    xu = u, and since for ANY y in S we have y = ua (for some a): xy = x(ua) = (xu)a = ua = y. thus x is a left-identity for S.

    similarly, there must be some z in S with uz = u, and thus (writing y = bu), yz = (bu)z = b(uz) = bu = y, so z is a right-identity for S.

    but then x = xz = z, so S possesses an identity element, which we can call e.

    so in THAT case, we have xu = e and uy = e for some x,y in S, so u is indeed a unit (and x = y: x = xe = x(uy) = (xu)y = ey = y).

    note that above, we only require that x→ux, x→xu be surjective to show that a right- or left-identity exists, uniqueness of this identity follows (for one of left- or right-) if one of those maps is injective as well (if S is finite, then surjective = bijective) (if x→ux is injective, then we have a unique right-identity, if x→xu is injective, we have a unique left-identity).

    i am unsure of how you would define a unit without the presence of a two-sided identity.
    Thanks from ymar
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    Re: a generalization of unit elements

    Thanks!
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