Lattice-ordered Groups

Abstract

In this chapter we present the most basic parts of the theory of lattice-ordered groups. Though our main concern will eventually be with abelian groups it is not appreciably harder to develop this material within the class of all groups. Moreover, the additional generality allows us to digress somewhat (if it is possible to digress before one begins) and to present some of the classical theorems in the subject. The fundamental interactions between the lattice structure and the group structure of a lattice-ordered group are dealt with first. Included are the characterization of the lattice structure in terms of the subsemigroup of positive elements as well as the elementary identities which result from the two structures. We then examine the morphisms in the category of lattice-ordered groups and the various kinds of subobjects. What is significant here is that the lattice of kernels in a lattice-ordered group is a distributive lattice and so is the corresponding lattice of subobjects that arises by dropping the normality requirement. These latter subobjects are precisely those for which the corresponding partition of cosets of the group is a lattice homomorphic image of the underlying lattice of the lattice-ordered group.