Coz-related and other special quotients in frames

Abstract

We study various quotient maps between frames which are defined by stipulating that they satisfy certain conditions on the cozero parts of their domains and codomains. By way of example, we mention that C-quotient and C -quotient maps (as defined by Ball and Walters- Wayland [7]) are typical of the types of homomorphisms we consider in the initial parts of the thesis. To be little more precise, we study uplifting quotient maps, C1- and C2-quotient maps and show that these quotient maps possess some properties akin to those of a C-quotient maps. The study also focuses on R - and G - quotient maps and show, amongst other things, that these quotient maps coincide with the well known C - quotient maps in mildly normal frames. We also study quasi-F frames and give a ring-theoretic characterization that L is quasi-F precisely when the ring RL is quasi-B´ezout. We also show that quasi-F frames are preserved and reflected by dense coz-onto R -quotient maps. We characterize normality and some of its weaker forms in terms of some of these quotient maps. Normality is characterized in terms of uplifting quotient maps, -normally separated frames in terms of C1-quotient maps and mild normality in terms of R - and G -quotient maps. Finally we define cozero complemented frames and show that they are preserved and reflected by dense z#- quotient maps. We end by giving ring-theoretic characterizations of these frames.