Unbounded Order Convergence in Ordered Vector Spaces

We consider an ordered vector space X. We define the net xα⊆X to be unbounded order convergent to x (denoted as xα⟶uox). This means that for every 0≤y∈X, there exists a net yβ (potentially over a different index set) such that yβ↓0, and for every β, there exists α0 such that ±xα−xu,yl⊆yβl whenever α≥α0. The emergence of a broader convergence, stemming from the recognition of more ordered vector spaces compared to lattice vector spaces, has prompted an expansion and broadening of discussions surrounding lattices to encompass additional spaces. We delve into studying the properties of this convergence and explore its relationships with other established order convergence. In every ordered vector space, we demonstrate that under certain conditions, every uo-convergent net implies uo-Cauchy, and vice versa. Let X be an order dense subspace of the directed ordered vector space Y. If J⊆Y is a uo-band in Y, then we establish that J∩X is a uo-band in X.