ABSTRACT
Transsymmetric spaces have been introduced by L. V. Sabinin as a far going generalization of Symmetric spaces and studied by L. L. Sabinina in Ph. D. dissertation. Despite of the geometric nature of the initial approach it has been clear that the whole theory may be reformulated in the terms of Smooth quasigroups and loops. This chapter outlines this algebraic approach. The triple Lie algebra, together with some unary operation, generated by endomorphism φ, and with some identities, constitute the proper infinitesimal object. Symmetric spaces possess some geometric properties which are not valid for the general transsymmetric spaces. This has motivated the notion of Perfect transsymmetric space. The chapter also outlines reductive spaces and left correct y-ts-loops.
