Duplication of linear algebras

Abstract

The process of duplication of a linear algebra was defined in an earlier paper, where its occurrence in the symbolism of genetics was pointed out. The definition will now be repeated with an amplification. Although for purpose of illustration it is applied to the algebra of complex numbers, duplication will seem of no special significance if attention is fixed on algebras with associative multiplication and unique division; for duplication generally destroys these properties. The results to be proved, however, show that it is significant in connection with various other conceptions which appeared in the discussion of genetic algebras; namely baric algebras and train algebras (defined in G.A.), also nilpotent algebras, linear transformation and direct multiplication of algebras.