A little more on coz-unique frames
A frame is \textit{coz-unique} in case its cozero part is the only regular sub-\(\sigma\)-frame generating it. Coz-unique frames were introduced and characterized by \textit{B. Banaschewski} and \textit{C. Gilmour} in [``Cozero bases of frames'', J. Pure Appl. Algebra 157, No. 1, 1--22 (2001; Zbl 0964.54020)]. In the paper under review, the author gives further characterizations of these frames. These results are extensions to frames of the following characterizations of absolutely \(z\)-embedded spaces obtained, respectively, by \textit{R. L. Blair} and \textit{A. W. Hager} [``Extensions of zero-sets and of real-valued functions'', Math. Z. 136, 41--52 (1974; Zbl 0264.54011)] and by \textit{A. W. Hager} and \textit{D. G. Johnson} [``A note on certain subalgebras of \(C(\mathfrak{X})\)'', Can. J. Math. 20, 389--393 (1968; Zbl 0162.26702)]: {\parindent=4mm \begin{itemize}\item[--] For every completely regular space \(X\) the following are equivalent: (1) \(X\) is absolutely \(z\)-embedded; (2) \(X\) is \(z\)-embedded wherever it is embedded; (3) \(X\) is either Lindelöf or almost compact; (4) If \(Z_1\) and \(Z_2\) are disjoint zero-sets of \(X\), either \(Z_1\) is compact or \(Z_2\) is Lindelöf; (5) If \(Z\) is a zero-set of \(X\), either \(Z\) is compact or \(X\setminus Z\) is Lindelöf. \item[--] Every embedding of a Tychonoff space \(X\) is a \(z\)-embedding if and only if the only realcompact space in which \(X\) is \(G_{\delta}\)-dense is the Hewitt realcompactification \(v X\). \end{itemize}} The spatial notion of \(z\)-embedded subspaces is captured in the point-free setting by coz-onto frame homomorphisms (that is, morphisms that map cozero parts of their domains onto cozero parts of their codomains). The paper ends with characterizations of these maps in terms of normal covers, along the lines of [\textit{R. A. Alò, L. Imler} and \textit{H. L. Shapiro}, Math. Ann. 188, 13--22 (1970; Zbl 0188.55202)].
scientific article; zbMATH DE number 5607534
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