Vector metrics with values in pre-Riesz spaces I: transfer of order convergence along vector lattice covers

Vector metric spaces in the sense of Çevik and Altun take values in a vector lattice, but their axioms use only the order of the value space. We develop the theory for vector metrics with values in an Archimedean pre-Riesz space E, i.e.\ an ordered vector space admitting a bipositive order dense embedding i into a vector lattice Y. We ask when an E-valued vector metric d and the vector-lattice-valued metric i∘ d have the same convergent and Cauchy nets and sequences. Relatively uniform convergence transfers along every cover. For nets, o₃-convergence transfers if and only if the cover has an intrinsic domination property. This property holds for pervasive covers, by a result of Kalauch and van Gaans, for uniformly dense covers and in finite dimensions; it fails for an example of Kalauch and van Gaans. Sequential convergence transfers under a countable domination property implied by uniform density, but not under pervasiveness. For uncountable $Γ$, the vector lattice R 1⊕ c₀₀(Γ) is pervasive in its Dedekind completion ^∞(Γ) without the transfer, while for countable $Γ$ the transfer holds; the difference is explained by order separability. In finite dimensions, with a closed generating cone, all these convergences coincide with norm convergence in every cover. For C¹[0,1]-valued integral metrics, positivity of the kernel on the diagonal forces all order convergences to coincide with L¹-convergence. For the Volterra kernel (t-s)₊, by contrast, L¹-convergence, relatively uniform convergence and order convergence are pairwise different, and we describe each explicitly.