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A mixed lattice vector space is a partially ordered vector space with twopartial orderings and certain lattice-type properties. In this paper we firstgive some fundamental results in mixed lattice groups, and then we investigatethe structure theory of mixed lattice vector spaces, which can be viewed as ageneralization of the theory of Riesz spaces. More specifically, we study theproperties of ideals and bands in mixed lattice spaces, and the related idea ofrepresenting a mixed lattice space as a direct sum of disjoint bands. Undercertain conditions, these decompositions can also be given in terms of orderprojections.

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