Quasi SV-frames and their properties

Abstract

Let P be a prime ideal of the ring of continuous real-valued functions on a completely regular frame L, i.e., L. We study many new results about the residue class domains L/P with an emphasis on determining when the ordered L/P is a valuation domain (i.e., when given any two non-zero elements of L/P , one divides the other). A prime ideal P of L is called a valuation prime ideal if L/P is a valuation domain. A frame L is called an SV-frame if every prime ideal of L is a valuation prime ideal. We introduce and study two new generalizations of the SV-frames. The first is that of a quasi SV-frame in which every real maximal ideal of L that is not a minimal prime ideal contains a non-maximal prime ideal P such that L/P is a valuation domain. In the second, we define a frame L to be an almost SV-frame if every maximal ideal of L contains a minimal valuation prime ideal. A point I ∈ Pt(βL) is called a special βF -point if O I = {δ ∈ L : coz δ ∈ I} is a valuation prime ideal of L. It is shown that I is a special βF -point if and only if the pseudo-prime ideals of L containing O I that are not primary form a chain under set inclusion.