The purpose of these notes is to give a rather complete presentation of the mathematical theory of algebras in genetics and to discuss in detail many applications to concrete genetic situations. Historically, the subject has its origin in several papers of Etherington in 1939- 1941. Fundamental contributions have been given by Schafer, Gonshor, Holgate, Reiers¢l, Heuch, and Abraham. At the moment there exist about forty papers in this field, one survey article by Monique Bertrand from 1966 based on four papers of Etherington, a paper by Schafer and Gonshor's first paper. Furthermore Ballonoff in the third section of his book "Genetics and Social Structure" has included four papers by Etherington and Reiers¢l's paper. Apparently a complete review, in par ticular one comprising more recent results was lacking, and it was difficult for students to enter this field of research. I started to write these notes in spring 1978. A first german version was finished at the end of that year. Further revision and translation required another year. I hope that the notes in their present state provide a reasonable review and that they will facilitate access to this field. I am especially grateful to Professor K. -P. Hadeler and Professor P. Holgate for reading the manuscript and giving essential comments to all versions of the text. I am also very grateful to Dr. I. Heuch for many discussions during and after his stay in TUbingen. I wish to thank Dr. V. M.

Inhalt
0. Introduction.- 1. Algebras in Genetics.- A. The occurence of algebras in genetics.- B. Algebras with genetic realization and baric algebras.- 2. Algebraic Preliminaries.- A. Non associative algebras definitions and notations.- B. Polynomials in non-associative powers.- C. The rank equation of an algebra.- 3. Train Algebras, Genetic Algebras and Special Train Algebras.- A. Train algebras.- B. Genetic algebras.- C. Special train algebras.- 4. Idempotents and Sequences of Powers in Train Algebras, Genetic Algebras and Algebras With Genetic Realization.- A. Idempotents in train algebras and genetic algebras.- B. Idempotents in algebras with genetic realization.- C. Sequences of principal and plenary powers in genetic algebras.- 5. The Non Commutative Case.- A. Train algebras, genetic algebras and special train algebras.- B. Idempotents and sequences of powers.- 6. Construction of New Algebras.- A. Linear combinations of algebras.- B. Tensor products.- C. Duplication.- D. Construction of new algebras by linear mappings.- 7. Applications I.- A. Additive segregation rates.- B. Partially coupled loci.- C. Autopolyploidy.- D. Overlapping and continuously overlapping generations.- 8. Applications II.- A. Asymmetric segregation rates.- B. Sex-linked inheritance.- C. Inbreeding systems.- 9. Concluding Remarks and Bernstein Algebras.- A. Open problems.- B. Bernstein algebras.- References.