ABSTRACT
During the last twenty six years in our work the foundations of Smooth quasigroups and Loops were established. Their notable relations with Geometry and Mathematical Physics were disclosed end explored. The infinitesimal theory of smooth loops was developed as well as the Bol loops-Bol algebras theory. This chapter discusses the theory of smooth M-loops and the infinitesimal theory of smooth left binary-Bol loops. Since the expansion of the law of composition into series in normal coordinates depends only on two elements, it is obvious that, like in the Bol case, the proper infinitesimal object will be a binary-ternary multilinear algebra.
