Real Algebraic and Analytic Geometry Preprint Server

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Submission: 2004, June 21.

Abstract:
We continue the study of a lattice-ordered ring $G(X)$, associated with the ring $C(X)$. Following \cite{HRW}, $X$ called $\RG$ when $G(X)=C(X_{\delta})$. An $\RG$-space must have a dense set of very weak $\P$-points. It must have a dense set of almost-$\P$-points if $X_{\delta}$ is Lindel\"of, or if the continuum hypothesis holds and $C(X)$ has small cardinality.
Space which are $\RG$ must have finite Krull dimension when taken with respect to the prime $z$-ideals of $C(X)$. There is a notion of regularity degree defined via the functions in $G(X)$. Pseudocompact spaces and metric spaces of finite regularity degree are characterized.

Mathematics Subject Classification (2000): 54G10, 46E25, 16E50.

Keywords and Phrases: P-space, almost-P space, prime z-ideal, RG-space, very weak $\P$-point.

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