  We study binary mixtures of ultra-cold atoms, confined to one dimension in an
optical lattice, with commensurate densities. Within a Luttinger liquid
description, which treats various mixtures on equal footing, we derive a system
of renormalization group equations at second order, from which we determine the
rich phase diagrams of these mixtures. These phases include charge/spin density
wave order, singlet and triplet pairing, polaron pairing, and a supersolid
phase. Various methods to detect our results experimentally are discussed.
