Representable, non-solvable varieties of lattice-ordered groups

The purpose of this paper is twofold. First, it is to study the structure of the space of infinite binary sequences under a specific relation and to prove that there are continuum many minimal sequences under this relation. Secondly, it is to expand the results of Holland and Medvedev (8). They proved that there exists a representable non-solvable variety of lattice-ordered groups associated with each binary sequence. Furthermore, they proved that contained within each variety associated with a minimal sequence there exists a cover of the Abelian variety. This paper will supply the results from the study of the space of binary sequences to prove that there are continuum many representable non-solvable covers of the Abelian variety and to prove that two of the varieties associated with minimal sequences are themselves covers (and do not merely contain covers) of the Abelian variety. Based on a series of conjectures, the generating groups of continuum many other covers as well as the structure of the class of representable non-solvable varieties will be explored.