One of the classic tools for studying the structure and classification of algebraic systems are derivative structures. The beginning of the study of such structures, which are clearly defined by a given system and contain an essential information about its general properties, dates back to the XIX century and associated with the works of the outstanding mathematician E. Galois. The most common derivative structures of algebraic systems traditionally include automorphism groups, endomorphism semigroups, congruence lattices and subalgebra lattices. This project is aimed at solving such a fundamental problem as the classification of endomorphism semigroups of algebraic systems up to an isomorphism, as well as to study the properties of free Loday’s structures (dimonoids, trioids). A special attention will be paid to the Plotkin’s problem about automorphisms of endomorphism semigroups of free algebras, the construction of relatively free algebras in the given varieties of structures of Loday, and the investigation of algebraic and combinatorial properties of endomorphism semigroups.