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?
it's very simple: your category will have one object, S. the morphisms of S will be the functions L_{z}:S-->S defined by L_{z}(x) = zx (we get one morphism for each element of S).
1. morphisms are composable: for each pair of morphisms, L_{z},L_{y} we have:
L_{z}oL_{y}(x) = L_{z}(L_{y}(x)) = L_{z}(yx) = z(yx) = (zy)x = L_{zy}(x) (since * is associative), for every x in S.
thus L_{z}oL_{y} = L_{zy}.
2. since 1 is an identity for S, we have the morphism L_{1}(x) = 1x = x. it is clear that L_{1}oL_{z} = L_{z} = L_{z}oL_{1}.
3. composition is associative (obvious).
that's it...now we have a category.