On the Nachbin compactification of products of totally ordered spaces (Q1902915)

The Nachbin (or Stone-Čech ordered) compactification is the largest \(T_2\)-ordered compactification \(\beta_0 X\) for a \(T_{3,5}\)- ordered space \(X\) (an ordered topological space which is ``completely regular ordered'' in the sense of Nachbin) [\textit{L. Nachbin}, Topology and order (1965; Zbl 0131.379)]. The main result of this paper deals with the product \(X\times Y\) of totally ordered spaces \(X\), \(Y\), where \(X\times Y\) has the product order and product topology. The following Theorem is proved: For totally ordered spaces \(X\), \(Y\) we have \(\beta_0 (X\times Y)= \beta_0 X\times \beta_0 Y\) if and only if the following condition is satisfied: If either \(X\) or \(Y\) contains an increasing (or decreasing) singularity of order \(\omega\), then the other space contains no strictly decreasing (or strictly increasing) sequence.