Lattice uniformities inducing unbounded convergence

Abstract

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