Applications of Duality in the Theory of Finitely Generated Lattice-Ordered Abelian Groups

In a previous paper by the author [3], duality theorems for finitely generated vector lattices and lattice-ordered Abelian groups are described. In particular, the category of finitely generated semi-simple vector lattices is shown to be equivalent to a geometrical category V whose objects are topologically closed cones in Euclidean space, and whose morphisms, called ll-maps\ form a special subclass of the class of piece wise homogeneous linear maps between such cones. Under this categorical duality, finitely generated projective vector lattices and closed polyhedral cones correspond; indeed, the category of finitely generated projective vector lattices is equivalent to the dual of a category whose objects are Euclidean closed polyhedral cones and whose morphisms consist of all piecewise homogeneous linear maps between such cones.