Singly generated frame extensions
As the title suggests, this paper investigates frames which arise from a given frame by adjoining a single element. More precisely, a frame \(M\) is a singly generated frame extension of a frame \(L\) if and only if \(L\) is a subframe of \(M\) and \(M\) is generated by \(L\) together with \(c\in M\). Such extensions are analyzed using congruences (equivalently nuclei) on the subframe \(P=\{(x,y)\mid x\leq y\}\) of \(L\times L\). This analysis includes a characterization of epi-extensions, that is, when the identity embedding \(L\to M\) is an epimorhism. The rest of the article considers frame theoretic separation properties, in particular the axiom of strong Hausdorffness (called ``separated'' here) due to \textit{J. R. Isbell} [Math. Scand. 31, 5-32 (1972; Zbl 0246.54028)]. It is first considered in the context of singly generated extensions, and then an appendix looks at it relative to more general extensions.
scientific article; zbMATH DE number 166149