Some aspects of dimension theory of frames
The authors study the concept of dimension in the context of frames. They start by recalling the standard definition of zero-dimensionality defined by declaring that a frame be generated by its complemented elements. Then they define a frame to be strongly zero-dimensional if every finite cover has a finite refinement of the same cardinality consisting of mutually disjoint elements. It should be noted that this is different from the usual definition, which says \(L\) is strongly zero-dimensional if \(\beta L\) is zero-dimensional. Indeed, in accordance with their definition, the authors show that a normal completely regular frame \(L\) is strongly zero-dimensional iff \(\beta L\) is strongly zero-dimensional. For a regular, a normal, and a completely regular frame \(L\), they define, respectively, the Menger-Urysohn dimension \(\operatorname{ind}L\), the Brouwer-Čech dimension \(\operatorname{Ind}L\) and the Čech-Lebesgue dimension \(\dim L\) analogously to spaces. The expected results extending the spatial ones, such as {\parindent=6,5mm \begin{itemize}\item[(a)]\(\operatorname{ind}L = 0\) iff \(L\) is zero-dimensional, \item[(b)] for normal regular \(L\), \(\operatorname{Ind}\beta L= \operatorname{ind}L\) and \(\dim L= \dim\beta L\), \item[(c)] \(\operatorname{ind}L\leq \operatorname{ind}L\), \end{itemize}} are established. The paper culminates with a look at connections between the various notions of dimension. Reviewer's comments: (a) Condition (3) in Theorem 3.11 should not be there as it is precisely the definition of strong zero-dimensionality stated as condition (1) in the theorem. (b) It is not necessary to provide a proof for Theorem 6.3 because this theorem follows from Theorem 6.4 (a proof of which is provided) since every regular Lindelöf frame is normal.
scientific article; zbMATH DE number 5501657