Abstract
The main aim of this paper is to provide a construction of the Banaschewski compactification of a zero-dimensional Hausdorff topological space as a structure space of a ring of ordered
field-valued continuous functions on the space, and thereby exhibit the independence of the construction from any
completeness axiom for an ordered field. In the process of describing this construction we have generalized the classical versions of M. H. Stone's theorem, the Banach-Stone theorem, and
the Gelfand-Kolmogoroff theorem. The paper is concluded with a conjecture of a split in the class of all
zero-dimensional but not strongly zero-dimensional Hausdorff topological spaces into three classes that are labeled by
inequalities between three compactifications of