Mixed Lattice Groups

Abstract

A mixed lattice group is a generalization of a lattice ordered group. The theory of mixed lattice semigroups dates back to the 1970s, but the corresponding theory for groups has been relatively unexplored. In this thesis we investigate the basic structure of mixed lattice groups, and study how some of the fundamental concepts in Riesz spaces and lattice ordered groups, such as the absolute value and other related ideas, can be extended to mixed lattice groups. We give a fundamental classification of mixed lattice groups based on their order properties. We define the generalized absolute values and derive various related identities and inequalities. We then introduce the concept of an ideal in a mixed lattice group and prove some basic results related to them. Using these ideas, we begin a study of homomorphisms in mixed lattice groups and their elementary properties. In this connection, we also investigate the quotient group construction, and show that under certain conditions, the quotient group will also be a mixed lattice group. Finally, we briefly consider topologies in mixed lattice groups and give a few sufficient conditions for a group topology to be compatible with the mixed lattice structure.