The function ring functors of pointfree topology revisited

Abstract

This paper establishes two new connections between the familiar function ring functor ${\mathfrak R}$ on the category ${\bf CRFrm}$ of completely regular frames and the category {\bf CR}${\mathbf \sigma}${\bf Frm} of completely regular $\sigma$-frames as well as their counterparts for the analogous functor ${\mathfrak Z}$ on the category {\bf ODFrm} of 0-dimensional frames, given by the integer-valued functions, and for the related functors ${\mathfrak R}^*$ and ${\mathfrak Z}^*$ corresponding to the bounded functions. Further it is shown that some familiar facts concerning these functors are simple consequences of the present results.

Keywords

• completely regular frames
• zero dimensional frames
• completely regular $sigma$-frames
• zero dimensional $sigma$-frames
• real-valued continuous functions and integer-valued continuous functions on frames