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Abstract
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.