We study variants of P-frames and associated rings, which can be viewed as natural
generalizations of the classical variants of P-spaces and associated rings. To be more
precise, we de ne quasi m-rings to be those rings in which every prime d-ideal is either
maximal or minimal. For a completely regular frame L, if the ring RL of real-valued
continuous functions of L is a quasi m-ring, we say L is a quasi cozero complemented
frame. These frames are less restricted than the cozero complemented frames. Using
these frames we study some properties of what are called quasi m-spaces, and observe
that the property of being a quasi m-space is inherited by cozero subspaces, dense z-
embedded subspaces, and regular-closed subspaces among normal quasi m-space.
M. Henriksen, J. Mart nez and R. G. Woods have de ned a Tychono space X to be a
quasi P-space in case every prime z-ideal of C(X) is either minimal or maximal. We call a
point I of L a quasi P-point if every prime z-ideal of RL contained in the maximal ideal
associated with I is either maximal or minimal. If all points of L are quasi P-points, we
say L is a quasi P-frame. This is a conservative de nition in the sense that X is a quasi
P-space if and only if the frame OX is a quasi P-frame. We characterize these frames
in terms of cozero elements, and, among cozero complemented frames, give a su cient
condition for a frame to be a quasi P-frame.
A Tychono space X is called a weak almost P-space if for every two zero-sets E and
F of X with IntE IntF, there is a nowhere dense zero-set H of X such that E F [H.
We present the pointfree version of weakly almost P-spaces. We de ne weakly regular
rings by a condition characterizing the rings C(X) for weak almost P-spaces X. We
show that a reduced f-ring is weakly regular if and only if every prime z-ideal in it which contains only zero-divisors is a d-ideal. We characterize the frames L for which the ring
RL of real-valued continuous functions on L is weakly regular.
We introduce the notions of boundary frames and boundary rings, and use them to
give another ring-theoretic characterization of boundary spaces. We show that X is a
boundary space if and only if C(X) is a boundary ring.
A Tychono space whose Stone- Cech compacti cation is a nite union of closed subspaces
each of which is an F-space is said to be nitely an F-space. Among normal spaces,
S. Larson gave a characterization of these spaces in terms of properties of function rings
C(X). By extending this notion to frames, we show that the normality restriction can
actually be dropped, even in spaces, and thus we sharpen Larson's result. / Mathematics / D. Phil. (Mathematics)
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:unisa/oai:uir.unisa.ac.za:10500/21795 |
| Date | 12 1900 |
| Creators | Nsayi, Jissy Nsonde |
| Contributors | Dube, T. A. |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Type | Thesis |
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