An epireflection ψ is constructed of the category KNArS of compact normal joinfit frames, with skeletal maps, in the subcategory SPArS consisting of strongly projectable KNArS-objects. The construction is achieved via a pushout in the category FrmS of frames with skeletal maps, and involves rather intimately the regular coreflection of the object to be reflected. Further, if the regular coreflection ρ is applied to the reflection map ψ A : A −→ ψ A one obtains the extension of ρ A to its absolute.Keywords Epireflection · Skeletal map · Coproduct · Pushout · Normal frame · Joinfitness · Strong projectability Mathematics Subject Classifications (2000) Primary 06D22; Secondary 18A20 · 18A30This article is the continuation of the work recorded in [20] and [15]; it continues in [16]. The first paper showed that the passage from a compact regular frame A to its absolute ε A is functorial, if one restricts to skeletal frame homomorphisms. In fact, the monoreflection ε is the functorial epicompletion on the category KRegS of all compact regular frames and skeletal frame maps. The goal of the research that went into [15] was to provide a bridge between [20] and this paper, by virtue of having gathered sufficient information about categories of archimedean frames and thereby having made this article easier to read. As indicated below, various

It is shown that, in KRegS, the category of compact regular frames with skeletal maps, the subcategory SPRegS, consisting of the frames in which every polar is complemented, coincides with the epicomplete objects in KRegS. Further, SPRegS is the least epireflective subcategory, and, indeed, the target of the monoreflection which assigns to a compact regular frame A, the ideal frame ε A of P A, the boolean algebra of polars of A. Mathematics Subject Classifications (2000) 06D22 · 18B15This research grew out of an interest in generalizing the related notions of (1) essential extensions of lattice-ordered groups and (2) irreducible surjections between compact Hausdorff spaces. As the title suggests, this is the first installment of two articles, dealing with the process of epicompletion in frames; the second (Martínez and Zenk, work in progress) will be concerned with coherent archimedean frames. The tools, which will be employed here and in Martínez and Zenk (work in progress), originate in the work on nuclear typings in [14], and the first steps in the direction of projectable hulls are already taken in [7]. The authors are grateful to Professor

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Nuclei which are defined over a class of frames are called nuclear typings. There is the dual notion of a spatial selector, and the relationship between nuclear typings and spatial selectors emanates from the duality between spatial frames and sober spaces. Especially interesting is the interplay between typings that are well-behaved with respect to certain frame quotients and selectors which similarly behave well in passage to closed sets. (2000): Primary 06D22, 54A05; Secondary 18B30, 18B35. Mathematics Subject Classification