Rings of continuous functions on σ-frames

This paper deals with the ℓ-rings RS of all real-valued continuous functions on a completely regular σ-frame. It shows that, in marked contrast with the situation for frames, any ℓ-ring homomorphism RS→RT results from a σ-frame homomorphism S→T. Further, it proves the analogue of this for integer-valued continuous functions and 0-dimensional σ-frames. In all, this demonstrates that the important classical difference between Alexandroff spaces and Tychonoff spaces with respect to the real-valued continuous functions carries over fully to the pointfree setting – indeed, it adds the integer-valued case which seems to be new in this context.