Directed ideals in partially ordered vector spaces

For a linear subspace II of a Riesz space there are various well-known properties that are equivalent to II being an ideal, such as II is a full Riesz subspace, II is a solid subspace, II is a Riesz subspace and the kernel of a positive linear map, II is the kernel of a Riesz homomorphism. Generalizations of these properties to partially ordered vector spaces are considered and their relations are investigated. It is shown that for directed subspaces all these generalizations are equivalent, just as in the case of Riesz spaces.