We show that the splicing algebras are an n-ary envelope for algebras of DNA recombination. We have constructed the basis for free algebra of the variety of the n-ary splicing algebras and found the defining identities for n-ary splicing operations. Using the relationship between algebras and its enveloping algebras, we have constructed the basis of the free algebra of the variety of n-ary algebras of DNA recombination. It is proved that every polynomial identity satisfied by n-ary DNA recombination, with no restriction on the degree, is consequence of n-ary commutativity and the special n-ary identity of the degree 3n-2.
We obtain a criterion, analogous to the Specht-Wever Lie criterion, for determining whether an element of n-ary free splicing algebra is an element of the algebra of DNA recombination. Using this criterion, it is proved that all n-ary algebras of DNA recombination are special by module of n-ary splicing algebra (analog of the Poincare-Birkhoff-Witt theorem).
The skew-symmetrization of n-ary splicing operation converts the splicing algebras into n-ary skew-symmetric algebras. In case n = 2, they are Lie algebras. It is proved that every polynomial identity of these algebras, with no restriction on the degree, is consequence of centrally metabelian identity.
In case n >2, they are nilpotent of index 3. The non-homologous recombination is formalized by the algebras of simplified insertions. It is shown that all identities of the algebra of simplified insertion follow from the right-symmetric identity. We construct an infinite series of relations in the algebra of simplified insertion which hold for the words of length n and are not consequences of the right-symmetric identity.
S. R. Sverchkov < SverchkovSR@yandex.ru >