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