Ordered compactifications of products of two totally ordered spaces
Let \(X\) and \(Y\) be totally ordered sets, and consider their totally ordered product \(X\times Y\). In this paper the authors study the semilattice of ordered compactifications of \(X\times Y\) smaller than \(\beta_0 X\times \beta_0 Y\). Here \(\beta_0 X\) is the Čech-Stone ordered- or Nachbin-compactification of \(X\). Similarly for \(Y\). For a totally ordered space \(Z\), let \(\sigma Z\) denote the smallest ordered compactification of \(X\). The authors prove results of the following nature. Suppose \(X\) and \(Y\) are totally ordered spaces having no simple singularities. Then \(\sigma X \times \sigma Y\) is the smallest ordered compactification of \(X\times Y\) less than \(\beta_0 X \times \beta_0 Y\). They are also able to characterize the semilattice of ordered compactifications in certain situations.
scientific article; zbMATH DE number 1463096