Compactness in Locally Convex Cones

We investigate notions of compactness with respect to the relative topologies of a locally convex cone. Since these topologies are generally not Hausdorff and the uniform structures induced by the upper and lower relative topologies are not symmetric, we introduce suitable generalizations for the concepts of precompactness and completeness, which nonetheless coincide with the usual ones in the symmetric case, that is in particular for the example of locally convex topological vector spaces. Locally compact cones comprise the topic for the final part of this paper.