Generalized absolute values, ideals and homomorphisms in 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 and vector spaces has been relatively unexplored. In this paper 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 and mixed lattice vector spaces. We also investigate ideals and study the properties of mixed lattice group homomorphisms and quotient groups. Most of the results in this paper have their analogues in the theory of Riesz spaces.

Ideals, bands and direct sum decompositions in mixed lattice vector spaces

“…In fact, a Riesz space is just a special case of a mixed lattice vector space, in which the two partial orderings coincide. Early studies of mixed lattice groups were done by Eriksson-Bique [5], [6], and more recently by the authors in [4]. The present paper is a continuation of the research that was commenced in [6], [5] and [4].…”

“…Early studies of mixed lattice groups were done by Eriksson-Bique [5], [6], and more recently by the authors in [4]. The present paper is a continuation of the research that was commenced in [6], [5] and [4].…”

“…This procedure turns S into a mixed lattice semigroup, provided that the conditions in the above definitions are satisfied. Many examples of mixed lattice structures are given in [4].…”

“…These are listed below, and will be used in this paper frequently. For more details concerning these rules and their proofs we refer to [4] and [6].…”

“…The concept of an ideal in mixed lattice groups and vector spaces was introduced in [4]. All the most relevant subspaces, including ideals, can be given in terms of their positive elements.…”