The dissertation is devoted to the study of functional equations and their solution on a set of binary invertible functions, the classification of quasigroup identities of the minimal length, the description of the corresponding varieties and their trusses. The relation of parastrophically primary equivalence is the main tool for the classification of functional equations. The parastrophic symmetry is the main law for the classification of identities and corresponding varieties.Some classes of quasigroups are considered, taking into account their groups of symmetries. The complete classification of group isotopes on the groups of their pa-rastrophic symmetries is given. The classification of linear isotopes of finite cyclic groups and isotopes of prime order groups is specified. The partition of the group isotopes in accordance with the strict parastrophic symmetry has been made, sets of all pairwise non-isomorphic group isotopes of prime order p (p>3) are highlighted and their power is calculated. A semi-symmetric isotope closure of a variety of the boolean groups and a variety of all groups are found, the necessary and sufficient conditions for their existence are obtained in terms of the canonical decomposition; the equivalence and the parastrophic equivalence of the identities which determine these varieties are defined. The relation between a variety of semi-symmetric quasigroups and a variety of semi-symmetric isotopic closures of boolean groups, abelian groups and all groups are found.Having defined the concept of the individual length of the equation, the individual type and the functional length, pure generalized non-trivial binary quasigroup functional equations up to the functional length of four of each of the possible individual types up to the parastrophically primary equivalence are classified. It is established that their number is exactly 1, 3, 4, 19 respectively for the equations of the length 1, 2, 3, 4. From each partition block of classification, a representative is selected and it is solved on a set of binary quasigroup operations. Namely, the complete classification of the generalized equations of the lengths 1, 2, 3, 4 up to the parastrophically primary equivalence is given; representatives of each block of the partition are found, the new ones of these are generalized equations of the type (4;0) of the length 2, of the type (5; 0) of the length 3 and of the types (6; 0; 0), (4; 2; 0) and (3; 3; 0) of the functional length 4. Representatives of all partition blocks that had not been solved before, more precisely, new types are solved; corollaries for quadratic, almost quadratic and anti-quadratic equations are given. In addition, four of the five known distributive-like functional equations without squares are solved.Having defined the notion of a generalized identity, we obtain two complete classi-fications of identities of the length 2 and 3 on quasigroups up to equivalence and the parastrophic equivalence and describe the distribution of the corresponding varieties of quasigroups into the trusses. It is proved that the identity of the length 2 defines 14 varieties, the identity of the length 3 defines 74 varieties, which respectively are classified into 6 and 20 trusses according to the parastrophic symmetry. Using the tools of the parastrophic symmetry, the varieties of solutions of generalized equations of the length 2 are found. All generalized identities of the length 2 and 3 up to equivalence and the parastrophic equivalence are described. The identities of the length 2 and 3 up to equivalence and parastrophic equivalence are found. The varieties of quasigroups which are determined by the obtained minimal identities are classified, they are analysed according to the groups of parastrophic symmetries. For each pair of parastrophic varieties examples of quasigroups which distinguish one variety from another in a single truss are found. Examples of quasigroups which distinguish one truss of varieties from another are found.The results of this dissertation study have of a theoretical nature. They can be used in the theory of functional equations, in the general theory of quasigroups, algebra, geometry, topology, mathematical analysis, combinatorics, cryptography, discrete ma-thematics and k-valued logic etc.
