Mixed Lattice Structures

In this thesis we develop the theory of mixed lattice structures, which are partially ordered algebraic systems (semigroups, groups or vector spaces) with two partial orderings. These two orderings are linked by mixed lattice operations which resemble the join and meet operations of a lattice but, unlike join and meet, the mixed lattice operations are neither commutative nor associative. A mixed lattice structure is a generalization of a lattice ordered structure in the sense that if the two partial orderings coincide then the resulting structure is a lattice.

We study several aspects of the mixed lattice theory. A generalization of the theory of vector lattices is developed by investigating how some of the fundamental concepts and the structure theory of vector lattices can be extended to mixed lattice structures. We introduce a fundamental classification of mixed lattice groups based on the properties of their orderings. We define the upper and lower parts of elements and derive various related identities and inequalities. Next we introduce the notions of ideals and bands in a mixed lattice space and prove several fundamental results related to these concepts. We then begin a study of the structure theory by investigating the notion of disjoint elements and the related decomposition of mixed lattice spaces into a direct sum of complemented bands.

In addition to the algebraic theory, we also study topologies on mixed lattice spaces. As our main result we present a characterization of compatible topologies on mixed lattice spaces and study the fundamental properties of such topologies. Applications to functional analysis and convex analysis are also presented. Finally, we introduce a generalization of a mixed lattice structure and show how such structure arises in convex analysis in the context of the cone projection problem.