Abstract
A lattice ordered group is a group which is also a lattice in which the group translations preserve the lattice structure. From the standpoint of group theory, the basic problem concerning lattice ordered groups is to determine conditions of a group theoretic nature which are either necessary or sufficient for a group to admit a lattice structure under which it becomes a lattice ordered group. We may describe this approach as the study of lattice orderable groups. In [3], Vinogradov and Vinogradov show that lattice orderability is not a local property, by constructing a group which is a union of finitely generated lattice orderable groups, but which is not itself lattice orderable. In this paper, we will show that lattice orderability is also not a residual property. We will construct a group which is a subdirect sum of lattice orderable groups but which is not itself lattice orderable. By way of contrast, it is known that total orderability of groups is both local (B. H. Neumann [2]) and residual (easily proved). The Scrimger 2-group S is the lattice ordered group consisting of all (x, y, z) Z 3 with group operation:
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