Symmetries on Partially Ordered Abelian Groups

Recently, Foulis (Foulis, D. J. (2003). Compressible groups, Mathematica Slovaca 53, 433–455.) characterized compressions on effect rings, which were introduced as a generalization of unital C*-algebras in the context of ordered abelian groups with order units. In the present paper, we characterize a class of symmetries on effect rings and show their relations to compressions. This characterization leads to a generalization of the notion of orthosymmetric orthoposets (Mayet, R., Pulmannová, S. (1994). Nearly orthosymmetric ortholattices and Hilbert spaces, Foundations of Physics 24, 1425–1437.) to symmetric ordered abelian groups with order units.