Extensions of the ring of continuous functions generated by the classical, rational, and regular rings of fractions as divisible hulls

The metaclassical extension generated by classical ring of quotients of the ring of continuous functions, the metarational extension generated by the rationally complete ring of quotients, and the metaregular extension generated by the regular ring of quotients, are considered along the lines of Fine–Gillman–Lambek. A new algebraic category of $c$-rings with refinement ($\equiv cr$-rings) is used to characterize them. Based on this the concept of a divisible $cr$-hull of step type is introduced. Parallel characterization are given of the metaclassical extension and the Riemann extension generated by Riemann-integrable functions, and also of the metarational and metaregular extensions and the Hausdorff–Sierpinski extension generated by semicontinuous functions.