On openness and surjectivity of lifted frame homomorphisms (Q989099)

It is well known that the functors \(\beta\) (Stone-Čech compactification), \(\lambda\) (universal Lindelöfication) and \(\upsilon\) (Hewitt realcompactification) on the category of completely regular frames lift any frame homomorphism \(h:L\rightarrow M\) to a unique frame homomorphism \(h^{\gamma}:\gamma L\rightarrow \gamma M\) such that \(\sigma_M\cdot h^{\gamma}=h\cdot \sigma_L\) (\(\gamma=\beta,\lambda,\upsilon\), respectively), where \(\sigma_K\) is the join map \(\gamma K\rightarrow K\) for any completely regular frame \(K\). However, openness of \(h\) is not preserved by \(\beta\); indeed, this already fails in the spatial case: it is possible for an open map \(f:X\rightarrow Y\) between Tychonoff spaces to be open whilst its Stone-Čech extension \(f^{\beta}:\beta X\rightarrow \beta Y\) is not open [\textit{S. Larson}, ``Images and open subspaces of SV spaces'', Commun. Algebra 36, No. 2, 352--364 (2008; Zbl 1147.54008)]. In this paper, the authors present a condition on the frame homomorphisms \(h\) such that maps satisfying the condition are open precisely when their lifts to the Stone-Čech compactifications are open. The condition is that the frame homomorphism \(h:L\rightarrow M\) should satisfy \(h^{\beta}\cdot r_L=r_M\cdot h\) (where \(r_K\) denotes the right adjoint of the join map \(\beta K\rightarrow K\), for any completely regular frame \(K\)). This is the point-free counterpart of what \textit{R. G. Woods} in [``Maps that characterize normality properties and pseudocompactness'', J. Lond. Math. Soc., II. Ser. 7, 453--461 (1974; Zbl 0271.54005)] calls an \(N\)-map, namely a continuous function \(f:X\rightarrow Y\) such that \(\text{cl}_{\beta X}f^{-1}[K]=(f^{\beta})^{-1}[\text{cl}_{\beta Y} K]\) for each closed set \(K\) in \(Y\). Accordingly, such a frame homomorphism \(h\) is called an \(N\)-map. Somewhat surprisingly, the same condition works for functors \(\lambda\) and \(\upsilon\): an \(N\)-map \(h\) is open iff \(h^\lambda\) is open, iff \(h^\upsilon\) is open. Furthermore, the same condition ensures that \(h^{\gamma}\) (for \(\gamma=\beta,\lambda\), or \(\upsilon\)) is nearly open iff \(h\) is nearly open. This is a special case of a more general phenomenon investigated in the last part of the paper.