Epicompletion in frames with skeletal maps. III: When maps are closed
In [\textit{J. Martínez} and \textit{E. R. Zenk}, ``Epicompletion in frames with skeletal maps. II: Compact normal joinfit frames'', Appl. Categ. Struct. 17, No.~5, 467--486 (2009; Zbl 1184.06005)], the authors constructed an epireflection \(\psi\) of the category \(\mathfrak{KNArS}\) of compact normal joinfit frames with skeletal maps, in the subcategory consisting of strongly projectable members. A frame \(L\) is joinfit if for any \(0<a\leq b\) in \(L\), there exists \(c\in L\) such that \(c<b\) and \(a\vee c=b\); and a frame homomorphism is skeletal if it maps dense elements to dense elements. By a strongly projectable frame is meant what frame theorists with topological leanings call extremally disconnected frames (Banaschewski prefers the moniker ``De Morgan frames''). A question which was left open is this article was whether \(\psi\) is a monoreflection. In the paper under review (which is the third in a series of papers investigating epicompletion in frames with skeletal maps), Martínez settles this question. This he achieves by showing that, for each \(A\) in \(\mathfrak{KNArS}\), the reflection map \(\psi_A\colon A\to\psi A\) is a closed homomorphism, whence, being dense, it is one-one. For an algebraic frame \(L\) in which the meet of two compact elements is compact, the map \(d\colon L\to L\) defined by \[ d(a)={\bigvee}\{c^{\perp\perp}\mid c \text{ is compact and } c\leq a\}, \] where \(\perp\) denotes pseudocomplement, is a nucleus. On the category of such frames, with skeletal maps, \(d\) defines an epireflection in the subcategory consisting of those objects for which \(dL=L\). In the second part of the paper the author establishes that the composite \(d\cdot\psi\) is an epireflection of the category of coherent normal joinfit frames with coherent (i.e. sends compact elements to compact elements) skeletal maps, in the subcategory of compact regular strongly projectable objects. The paper concludes with a number of open questions and work points. Readers who are familiar with the author's works know his very lucid and clear style of writing. This article keeps with that tradition.
scientific article; zbMATH DE number 5886915
scientific article