Hidden semi-group structures in crystals

The ordering and Mbius method has been applied to a variety of inverse problems in physics, such as the inverse blackbody, heat capacity and Fermi system problems. Recent progress in the application of this methodology to the inverse cohesion problem for inter\|atomic potentials in bulk materials and to the inverse adhesion problem for inter\|atomic potentials across interfaces is reviewed. The key step in this methodology is to explore the semi\|group structure in the system under consideration.