For a standard graded algebra $R$, we consider embeddings of the the poset of
Hilbert functions of quotients of $R$ into the poset of ideals of $R$, as a way
of classification of Hilbert functions. There are examples of rings for which
such embeddings do not exist. We describe how the embedding can be lifted to
certain ring extensions, which is then used in the case of polarization and
distraction. A version of a theorem of Clements--Lindstr\"om is proved. We
exhibit a condition on the embedding that ensures that the classification of
Hilbert functions is obtained with images of lexicographic segment ideals.