On the Function Ring Functor in Pointfree Topology

摘要

It is shown that the familiar existence of a left adjoint to the functor from the category of frames to the category of archimedean commutative f-rings with unit provided by the rings of pointfree continuous real-valued functions is already a consequence of a minimal amount of entirely obvious information, and this is then used to obtain unexpectedly simple proofs for a number of results concerning these function rings, along with their counterparts for the rings of integer-valued continuous functions in this setting. In addition, two different concrete descriptions are given for the left adjoint in question, one in terms of generators and relations motivated by the propositional theory of ?-ring homomorphisms into R, and the other based on a new notion of support specific to f-rings.