Riesz completions, functional representations, and anti-lattices

We show that the Riesz completion of an Archimedean partially ordered vector space with unit can be represented as a norm dense Riesz subspace of the smallest functional representation of This yields a convenient way to find the Riesz completion. To illustrate the method, the Riesz completions of spaces ordered by Lorentz cones, cones of symmetric positive semi-definite matrices, and polyhedral cones are determined. We use the representation to analyse the existence of non-trivial disjoint elements and link the absence of such elements to the notion of anti-lattice. One of the results is a geometric condition on the dual cone of a finite dimensional partially ordered vector space that ensures that is an anti-lattice.