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.
Highlights
The usual supremum and infimum of two elements is replaced by unsymmetrical mixed envelopes which are formed with respect to the two partial orderings
Since we are considering a generalization of lattice ordered groups and Riesz spaces, it is natural to begin our study by considering some of the most fundamental concepts in these structures, such as the absolute value and ideals, and examine how these concepts could be carried over to the theory of mixed lattice structures
We study the kernel of a mixed lattice homomorphism and show that the kernel has similar ideal properties as the kernels of Riesz homomorphisms between two Riesz spaces (Theorem 5.9)
Summary
We begin by stating the definitions of the basic structures. Let (S, +, ≤) be a positive partially ordered Abelian semigroup with zero element. We define a mixed lattice vector space by introducing another partial ordering and requiring that the mixed envelopes exist for every pair of elements. (V, ≤, ) is a mixed lattice group (a mixed lattice vector space) which is quasi-regular, but not regular since the specific cone C2 is not generating This example shows that a mixed lattice group need not be a lattice with respect to either partial ordering. Example 2.22 The set R of real numbers is a mixed lattice group with ≤ as the usual order and specific order defined by x y if y − x ≥ 0 and y − x ∈ Z. In an ordered vector space the positive cone is always convex, so this raises the following question: Does there exist a pre-regular mixed lattice vector space that is not quasi-regular? It turns out that more interesting results can be obtained under the pre-regularity assumption, and in later sections we will mostly focus on the pre-regular case
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