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Math Help - Define a category from a monoid

  1. #1
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    iaoalst
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    Define a category from a monoid

    I have the semigroup (S,*) with the identity 1. For ∀ x,y ∈ S
    and the set S(x,y) = {z ∈ S: x = zy }

    How can I show that this can be "expanded"/transformed/extended into a category structure on S?
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  2. #2
    MHF Contributor

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    Re: Define a category from a monoid

    it's very simple: your category will have one object, S. the morphisms of S will be the functions Lz:S-->S defined by Lz(x) = zx (we get one morphism for each element of S).

    1. morphisms are composable: for each pair of morphisms, Lz,Ly we have:

    LzoLy(x) = Lz(Ly(x)) = Lz(yx) = z(yx) = (zy)x = Lzy(x) (since * is associative), for every x in S.

    thus LzoLy = Lzy.

    2. since 1 is an identity for S, we have the morphism L1(x) = 1x = x. it is clear that L1oLz = Lz = LzoL1.

    3. composition is associative (obvious).

    that's it...now we have a category.
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