Special Valued L-groups

Special elements and values have always been of interest in the study of lattice-ordered groups, arising naturally from totally-ordered groups and lexicographic extensions. Much work has been done recently with the class of lattice-ordered groups whose root system of regular subgroups has a plenary subset of special values. We call such l-groups special valued. In this paper, we first show that several familiar structures of l-groups, namely polars, minimal prime subgroups, and the lex kernel, are recognizable from the lattice and the identity; that is, knowing which element of the lattice is the group identity, we can pick out in the lattice all the dements of polars, minimal primes, and the lex kernel. This then leads to an easy proof that special elements can be recognized from the lattice and the identity. We then prove several results about the class S of special-valued l-groups. We give a simple and direct proof that S is closed with respect to joins of convex l-subgroups, incidentally giving a direct proof that S is a quasi torsion class. This proof is then used to show that the special-valued and finite-valued kernels of l-groups are recognizable from the lattice and the identity. We show also that the lateral completion of a special-valued l-group is special-valued and is an a*-extension of the original l-group. Our most important result is that the lateral completion of a completely-distributive normal-valued l-group is special-valued. This lends itself easily to a new and similar proof of Ball, Conrad, and Darnel's result that every normal-valued l-group can be l-embedded into a special-valued l-group. Readers familiar with the impact of the Conrad-Harvey-Holland Theorem on abelian l-groups will recognize the importance of the last theorem to the study of the class of normal-valued l-groups and to the study of proper varieties of l-groups, all of which are normal valued.