Unbounded Order Convergence in Ordered Vector Spaces

We consider an ordered vector space . We define the net to be unbounded order convergent to (denoted as ). This means that for every , there exists a net (potentially over a different index set) such that , and for every , there exists such that whenever . 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 -convergent net implies -Cauchy, and vice versa. Let be an order dense subspace of the directed ordered vector space . If is a -band in , then we establish that is a -band in .