The question to
which this study addresses itself is the following: given a completely regular
space 𝒳, is the Dedekind completion of C(𝒳) isomorphic to C(𝒴) for some
space 𝒴? Here, C(𝒳) denotes the ring of continuous realvalued functions on
𝒳 under pointwise order. Affirmative answers were provided by Dilworth
for the class of compact spaces in 1950 and by Weinberg for the class of
countably paracompact and normal spaces in 1960. It remained an open question
whether there were any spaces for which a negative answer held. In this
paper, we provide a necessary and sufficient condition that the Dedekind
completion of C(𝒳), for 𝒳 a realcompact space, be isomorphic to C(𝒴) for some
𝒴. Using this, we are able to provide an example of a space 𝒳 for which
the Dedekind completion of C(𝒳) is not isomorphic to C(𝒴) for any space
𝒴.
