Distributive ℓ-pregroups: generation and decidability

We show that every distributive lattice-ordered pregroup can be embedded into a functional algebra over an integral chain, thus improving the existing Cayley/Holland-style embedding theorem. We use this result to show that the variety of all distributive lattice-ordered pregroups is generated by the functional algebra on the integers. Finally, we show that the equational theory of the variety is decidable.