This monograph develops the theory of pre-Riesz spaces which are the partially ordered vector
spaces that embed order densely into Riesz spaces. Concepts from Riesz space theory such as
disjointness ideals and bands are extended to pre-Riesz spaces. The analysis revolves around
embedding techniques including the Riesz completion and the functional representation. In the
same spirit norms and topologies on a pre-Riesz space and their extensions to the Riesz
completion are examined. The generalized concepts are used to investigate disjointness
preserving operators on pre-Riesz spaces and related notions. The monograph presents recent
results as well as being an accessible introduction to the theory of partially ordered vector
spaces and positive operators. Contents A primer on ordered vector spaces Embeddings covers
and completions Seminorms on pre-Riesz spaces Disjointness bands and ideals in pre-Riesz
spaces Operators on pre-Riesz spaces
