The ring of real-valued continuous functions on a completely regular frame L is denoted by
RL. As usual, βL denotes the Stone-Cech compactification of ˇ L. In the thesis we study ideals
of RL induced by sublocales of βL. We revisit the notion of purity in this ring and use it to
characterize basically disconnected frames. The socle of the ring RL is characterized as an ideal
induced by the sublocale of βL which is the join of all nowhere dense sublocales of βL.
A localic map f : L → M induces a ring homomorphism Rh: RM → RL by composition, where
h: M → L is the left adjoint of f. We explore how the sublocale-induced ideals travel along the
ring homomorphism Rh, to and fro, via expansion and contraction, respectively.
The socle of a ring is the sum of its minimal ideals. In the literature, the socle of RL has been
characterized in terms of atoms. Since atoms do not always exist in frames, it is better to
express the socle in terms of entities that exist in every frame. In the thesis we characterize the
socle as one of the types of ideals induced by sublocales.
A classical operator invented by Gillman, Henriksen and Jerison in 1954 is used to create a
homomorphism of quantales. The frames in which every cozero element is complemented (they
are called P-frames) are characterized in terms of some properties of this quantale homomorphism.
Also characterized within the category of quantales are localic analogues of the continuous maps
of R.G. Woods that characterize normality in the category of Tychonoff spaces. / Mathematical Sciences / Ph. D. (Mathematics)
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:unisa/oai:uir.unisa.ac.za:10500/27818 |
| Date | 08 1900 |
| Creators | Stephen, Dorca Nyamusi |
| Contributors | Dube, T. A. |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | 1 online resource (viii, 101 leaves), application/pdf |
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