The left adjoint of Spec from a category of lattice-ordered groups

Let us write ℓGuf for the category whose objects are lattice-ordered abelian groups (l-groups for short) with a strong unit and finite prime spectrum endowed with a collection of Archimedean elements, one for each prime l-ideal, which satisfy certain properties, and whose arrows are l-homomorphisms with additional structure. In this paper we show that a functor which assigns to each object (A,uˆ)∈ℓGuf the prime spectrum of A, and to each arrow f:(A,uˆ)→(B,vˆ)∈ℓGuf the naturally induced p-morphism, has a left adjoint.