  We present the lattice structure of Feynman diagram renormalization in
physical QFTs from the viewpoint of Dyson-Schwinger-Equations and the core Hopf
algebra of Feynman diagrams. The lattice structure encapsules the nestedness of
diagrams. This structure can be used to give explicit expressions for the
counterterms in zero-dimensional QFTs using the lattice-Moebius function.
Different applications for the tadpole-free quotient, in which all appearing
elements correspond to semimodular lattices, are discussed.
