Special-valued \(\ell\)-groups
Special elements and special 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 \(\ell\)- groups special-valued. In this paper, we first show that several familiar structures, namely polars, minimal prime subgroups, and the lex kernel, are recognizable from the lattice and the identity. This then leads to an easy proof that special elements can also be recognized from the lattice and the identity. We then give a simple and direct proof that S, the class of special-valued \(\ell\)-groups, is closed with respect to joins of convex \(\ell\)-subgroups, incidentally giving a direct proof that S is a quasitorsion class. This proof is then used to show that the special- valued and finite-valued kernels of \(\ell\)-groups are recognizable from the lattice and the identity. We also show that the lateral completion of a special-valued \(\ell\)-group is special-valued and is an \(a^*\)- extension of the original \(\ell\)-group. Our most important result is that the lateral completion of a completely distributive normal-valued \(\ell\)-group is special-valued. This lends itself easily to a new and simple proof of a result by Ball, Conrad, and Darnel that generalizes the Conrad-Harvey-Holland Theorem, namely, that every normal-valued \(\ell\)-group can be v-embedded into a special-valued \(\ell\)-group.
scientific article; zbMATH DE number 3931085
scientific article