Free Banach lattices over pre-ordered Banach spaces

We study free Banach lattices over pre-ordered Banach spaces in the category of Banach lattices of a given convexity type. These generalise the free Banach lattices under convexity conditions over Banach spaces in the literature. Their existence is shown from the existence of free vector lattices over pre-ordered vector spaces, which are also investigated. We determine when the positive contraction from the pre-ordered Banach space into the free Banach lattice is injective or bipositive, and when it has closed range. It is a bipositive embedding with closed range if and only if the positive wedge of the space is a closed normal cone. Even for a Banach lattice it can be non-isometric. By analysing the norm of the free $p$-convex Banach lattice with convexity constant 1 over a pre-ordered Banach space, it becomes clear that it can be realised as a function lattice on the positive part of the dual unit ball. This generalises the known realisation for a free Banach lattice of that type over a Banach space. As a preparation for this analysis of the norm, characterisations of $p$-convex Banach lattices in terms of vector lattice homomorphisms into $\mathrm{L}_p(\mu)$-spaces for probability measures $\mu$ are given.