Table of contents

CONTENTS Introduction Chapter I. The spaces and their properties § 1. The spaces § 2. The spaces § 3. The spaces Chapter II. Embedding theorems § 4. The concept of embedding theorems § 5. Embedding theorems for functions defined on the whole space § 6. Restrictions of functions belonging to the space § 7. Embedding theorems for a bounded domain § 8. Complete continuity of the embedding and restriction operators § 9. The embedding of in . Examples Appendix I. Embedding theorems for spaces of vector functions § 10. Weight matrices and their properties § 11. The spaces and and their properties § 12. Embedding theorems for the spaces and Appendix II § 13. The spaces References

CONTENTS Introduction Definitions, notation, and results from the theory of quasigroups Chapter I. Identities in systems of quasigroups § 1. Definitions and examples § 2. General identities § 3. Conjugate identities § 4. Some remarks on hyperidentities § 5. Generalized identities Chapter II. Functional equations on quasigroups. Systems of quasigroups with generalized associative, medial and transitive laws § 1. General associative law § 2. General medial law § 3. General transitive equation § 4. General distributive law § 5. General Stein identity. Orthogonal quasigroups Chapter III. Systems of quasigroups with generalized distributive law § 1. The connection of D-systems with commutative Moufang loops § 2. Maximal D-systems § 3. Commutative D-systems § 4. Properties of the transition permutations § 5. A criterion for generalized distributivity § 6. Homogeneity of the operations in a D-system Chapter IV. Systems of quasigroups with generalized Stein identity § 1. Definitions § 2. Auxiliary results § 3. Idempotence of the operations in D-systems § 4. Interdependence of n and s § 5. Connection between S-systems and D-systems §6. Complete S-systems References