Epicompletion in frames with skeletal maps. IV: \(\ast\)-regular frames
In Part I of this series of papers [Appl. Categ. Struct. 16, No. 4, 521--533 (2008; Zbl 1156.06004)], the author and \textit{E. R. Zenk} started an investigation into epicompletion in various categories of compact frames with skeletal maps. The paper under review is the fourth installment in this series of investigations. In the second installment [the author and \textit{E. R. Zenk}, ibid. 17, No. 5, 467--486 (2009; Zbl 1184.06005)], an epireflection, denoted \(\psi\), was constructed. This was an epireflection of the category of compact normal joinfit frames in the subcategory consisting of strongly projectable (also known as extremely disconnected) frames. In the third contribution in this anthology [the author, ibid. 19, No. 2, 489--504 (2011; Zbl 1223.06010)], one of the main achievements was the demonstration that the epireflection \(\psi\) is actually a monoreflection, and that the reflection map is a closed map. In this article, the author continues the investigation. He introduces \(\ast\)-regular frames -- a notion which broadens the domain of the monoreflection \(\psi\). These are frames for which the inclusion map from the regular coreflection of a frame into the frame is \(\ast\)-dense; meaning that, whenever the right adjoint of the inclusion map sends an element to the bottom, then the element is the bottom. The usual saturation nucleus is shown to be a reflection with respect to weakly closed maps, in the subcategory of subfit objects. Furthermore, the saturation nucleus commutes with \(\psi\) on compact normal joinfit frames with skeletal maps which are weakly closed. The paper closes with an illustration of the scope of the material developed in the earlier parts. This includes examples of frames that satisfy some properties, but not others. For instance, there is an example (which the author attributes to Warren McGovern) of a coherent \(\ast\)-regular frame which is not normal. Building on this example, using saturation and the monoreflection \(\psi\), the author manufactures a compact, subfit, strongly projectable, \(\ast\)-regular frame which is not regular.
scientific article; zbMATH DE number 6087663
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