Compressions on partially ordered abelian groups

If $A$ is a C*-algebra and $p\in A$ is a self-adjoint idempotent, the mapping $a\mapsto pap$ is called a compression on $A$. We introduce effect-ordered rings as generalizations of unital C*-algebras and characterize compressions on these rings. The resulting characterization leads to a generalization of the notion of compression on partially ordered abelian groups with order units.