On The Function Rings of Pointfree Topology

Abstract

The purpose of this note is to compare the rings of continuous functions, integer-valued or real-valued, in pointfree topology with those in classical topology. To this end, it first characterizes the Boolean frames (= complete Boolean algebras) whose function rings are isomorphic to a classical one and then employs this to exhibit a large class of frames for which the functions rings are not of this kind. An interesting feature of the considerations involved here is the use made of nonmeasurable cardinals. In addition, the integer-valued function rings for Boolean frames are described in terms of internal lattice-ordered ring properties. As is familiar, pointfree topology - that is, the setting of frames - shares with classical topology the fact that each basic entity (spaces in one case, frames in the other) has associated with it the ring of its real-valued continuous functions, and this in such a way that the correspondence for frames extends that for spaces. To be precise, if RL is the ring associated with a frame L and OX the frame of open sets of a space X then the classical function ring C(X) is naturally isomorphic to R(OX). It may be added here that the correspondence X 7! OX eects a full dual embedding into the category of frames of the category of Tychono spaces - the natural context for considering the rings C(X). Now, given that there is a large supply of non-spatial frames, that is, frames not isomorphic to any OX, the correspondence L 7! RL is certainly a proper extension of the correspondence X 7! C(X), via the intervening X 7! OX. That, however, does not a priori exclude the possibility that every RL might be isomorphic to some C(X) but in fact this is not the case, and one of the purposes of this note is to describe a method of verifying this. There are other ways of doing this, as will be discussed later; the present approach is to provide first a characterization of the Boolean frames L with RL isomorphic to some C(X) and then to show that this excludes all non-atomic L of nonmeasurable cardinal. Actually, it turns out to be convenient to consider these matters first for the