Riesz completions, functional representations and anti-lattices

We show that the Riesz completion of an Archimedean partially or- dered vector space X with unit can be represented as a norm dense Riesz subspace of the smallest functional representation of X. This yields a con- venient way to compute the Riesz completion. To illustrate the method, the Riesz completions of spaces ordered by Lorentz cones, cones of sym- metric positive semi-definite matrices, and polyhedral cones are deter- mined. We use the representation to investigate the existence of non- trivial disjoint elements and link the absence of such elements to the no- tion of anti-lattice. One of the results is a geometric condition on the dual cone of a finite dimensional partially ordered vector space X that ensures that X is an anti-lattice.