Functorial maximal spectra

Maximal spectra, of one kind or another, are usually not expected to be functorial, in notable contrast with the corresponding prime spectra, but there are cases in which the particular nature of the structures involved make them so. Here, we consider this in the context of compact normal frames, establishing the more fundamental functoriality of their saturation quotients which then readily implies that of their maximal spectrum, and showing this to be the common root of results concerning f-rings (J.R. Isbell, J. London Math. Soc. 40 (1965) 63), Gelfand rings (C.J. Mulvey, J. Algebra 56 (1979) 499), and the maximal ideals of C(X) (I.M. Gelfand, A.N. Kolmogoroff, Doklady Acad. Nauk USSR 22 (1939) 11).