Motivation for the study of partially ordered abelian groups has come from many different parts of mathematics, for mathematical systems with compatible order and additive (or linear) strlk'ctures are quite common. This is particularly evident in functional analysis, where spaces of various kinds of real-valued functions provide impetus for investigating partially ordered real vector spaces. In the past decade, the observation that a Grothendieck group (such as KD of a ring or algebra) often possesses a natural partially ordered abelian group structure has led to new directions of investigation, whose goals have been to develop structure theories for certain types of partially ordered abelian groups to the point where effective application to various Grothendieck groups is possible. Such recent developments in the area of partially ordered abelian groups are the subject of this note. We present a sketch of the construction of Grothendieck groups as abelian groups equipped with pre-orderings tha
