Saturation, Yosida Covers and Epicompleteness in Compact Normal Frames

Abstract

In this article the frame-theoretic account of what is archimedean for order-algebraists, and semisimple for people who study commutative rings, deepens with the introduction of ${\mathcal{J}}$-frames: compact normal frames that are join-generated by their saturated elements. Yosida frames are examples of these. In the category of ${\mathcal{J}}$-frames with suitable skeletal morphisms, the strongly projectable frames are epicomplete, and thereby it is proved that the monoreflection in strongly projectable frames is the largest such. That is news, because it settles a problem that had occupied the first-named author for over five years. In compact normal Yosida frames the compact elements are saturated. When the reverse is true one gets the perfectly saturated frames: the frames of ideals Idl(E), when E is a compact regular frame. The assignment E↦Idl(E) is then a functorial equivalence from compact regular frames to perfectly saturated frames, and the inverse equivalence is the saturation quotient. Inevitable are the Yosida covers (of a ${\mathcal{J}}$-frame L): coherent, normal Yosida frames of the form Idl(F), with F ranging over certain bounded sublattices of the saturation SL of L. These Yosida frames cover L in the sense that each maps onto L densely and codensely. Modulo an equivalence, the Yosida covers of L form a poset with a top ${\mathcal{Y}} L$, the latter being characterized as the only Yosida cover which is perfectly saturated. Viewed correctly, these Yosida covers provide, in a categorical setting, another (point-free) look at earlier accounts of coherent normal Yosida frames.