## Time ordering operator

As I understand it, the time ordering operator works as follows (for $$t<0$$):
$$T c^\dagger(t) c(0) = -c(0) c^\dagger(t)$$ for fermions and
$$T c^\dagger(t) c(0) = c(0) c^\dagger(t)$$ for bosons.

Now suppose instead of these creation/annihilation operators, I had a more general commutation relation, ie $$[d,d^\dagger] = S$$, how does the time ordering operator behave?

Edit: After rereading that, I should be more specific. I'm trying to formulate DMFT equations in a non-orthogonal basis. So the creation/annihilation operators anti-commute to give [d_a,d_b^\dagger]_+ = S_{a,b}. The Green's function is usually defined as $$G(\tau) = \langle T c(\tau) c^\dagger(0) \rangle$$ and I need to understand exactly what the time ordering operator does, but I'm not even totally sure how its really defined, because as I understand it above, it works differently for fermions and bosons.
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