Lattice uniformities inducing unbounded convergence

A net (xγ)γ∈Γ in a locally solid Riesz space (X,τ) is said to be unbounded τ-convergent x if |xγ−x|∧u⟶τ0 for all u∈X+. We recall that there linear topology uτ on X such τ-convergence coincides with uτ-convergence. It turns out characterised as the weakest which τ order bounded subsets. this motivation we introduce, uniform lattice (L,u), uniformity u⁎ L u subsets of L. shown induced by (X,τ), then u⁎-topology uτ. This allows comparing results paper earlier τ-convergence. will seen despite fact setup lattices most machinery used techniques [24] lacking, concept ‘unbounded convergence’ well fittingly generalizes lattices. shall also answer Questions 3.3, 5.10 and Question 18.51 [22].