  The conception of multi-alphabetical genetics is represented. Matrix forms of
the representation of the multi-level system of molecular-genetic alphabets
have revealed algebraic properties of this system. These properties are
connected with well-known notions of dyadic groups and dyadic-shift matrices.
Matrix genetics shows relations of the genetic alphabets with some types of
hypercomplex numbers including dual numbers and bicomplex numbers together with
their extensions. A possibility of new approach is mentioned to simulate
genetically inherited phenomena of biological spirals and phyllotaxis laws on
the base of screw theory and Fibonacci matrices. Dyadic trees for sub-sets of
triplets of the whole human genome are constructed. A new notion is put forward
about square matrices with internal complementarities on the base of genetic
matrices. Initial results of the study of such matrices are described. Our
results testify that living matter possesses a profound algebraic essence. They
show new promising ways to develop algebraic biology.
