What Remains the Same in Order Convergence Types

In this paper, we examine what remains the same between order convergence and unbounded order convergence, as well as between unbounded order continuity and strongly unbounded order continuity. In [1], Gao et al. proved that a sublattice of a Riesz space is order closed if and only if it is unbounded order closed. It is shown that σ-ideals and unbounded σ-ideals are the same. Additionally, it is established that injective band operators are unbounded order continuous, while bijective order bounded disjoint preserving operators are order continuous. Let G be an order dense majorizing Riesz subspace of a Riesz space E, and let F be a Dedekind complete Riesz space. In reference [2], the question is posed: If T:G→F is a positive strongly unbounded order continuous operator, does T have a unique positive strongly unbounded order continuous extension to all of E? We prove that this problem has a positive answer whenever G is suo-convergence reducing of E, namely, if xα→suo0 in E then xα→uo0 in G for any net (xα) in G.