Constructing Banaschewski compactification without Dedekind completeness axiom

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 X , namely, the Stone-ÄŒech compactification β X , the Banaschewski compactification β 0 X , and the structure space 𝔐 X , F of the lattice-ordered commutative ring â„­ ( X , F ) of all continuous functions on X taking values in the ordered field F , equipped with its order topology. Some open problems are also stated.