Abstract:
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 $\sigma$-ideals and unbounded $\sigma$-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 $T : G\rightarrow F$ 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 $ x_\alpha \overset{suo}{\rightarrow} 0$ in $E$ then $x_\alpha \overset{uo}{\rightarrow} 0$ in $G$ for any net $(x_\alpha)$ in $G$.
Key words:unbounded order convergent, unbounded order closed ideal, unbounded order continuous operator, strongly unbounded order continuous operator.
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