Lattice of Definability in the Order of Rational Numbers

A lattice of definability subspaces in the order of rational numbers is described. It is proved that this lattice consists of five subspaces defined in the paper that are generated by the following relations: “equality,” “less,” “between,” “cycle,” and “linkage.” For each of the subspaces, its width (the minimum number of arguments of a generating relation) is found and a convenient description of the automorphism group is given. Although the structure of this lattice was known previously, the proof in the paper is of syntactic nature and avoids the use of a group-theoretical method.