On unbounded order convergence (Q1978851)

Let \(E\) be a Riesz space. A net \((a_{\alpha })\) is said to be unboundedly convergent to \(a\in E\) if for every \(b,c\in E\), \(b\leq c, (a_{\alpha }\wedge c)\vee b\) is order convergent to \((a\wedge c)\vee b\). Two equivalent conditions in Dedekind complete spaces with a week order unit are given, e.g. \(|a_{\alpha } - a |\wedge 1 \to 0\) in order. \(E\) is said to be laterally \(\sigma \)-complete if every countable \(A\subset E\) has a supremum provided \(a\wedge b = 0\) for any \(a,b\in A\), \(a\neq b\). The author proves the Nakano statement : If \(E\) is Dedekind \(\sigma \)-complete and laterally \(\sigma \)-complete then any unboundedly convergent sequence is order bounded.