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On $$k$$th-order Bernstein algebras and stability at the $$k+1$$ generation in polyploids. (English) Zbl 0726.17040
Arising out of a problem raised by S. Bernstein [C. R. Acad. Sci. Paris 177, 581–584 (1923)], so-called Bernstein algebras have been studied in the last two decades. In the present paper the generalisation to $$k$$th-order Bernstein algebras (Bernstein algebras come from $$k=1)$$ are studied, having been introduced by V. M. Abraham [Proc. Lond. Math. Soc., III. Ser. 40, 346-363 (1980; Zbl 0388.17007)]. Thus, let $$A$$ be a commutative nonassociative algebra over a field $$K$$, and let $$x\in A$$. Define $$x^{[1]}=x$$, $$x^{[2]}=x^ 2,...$$, $$x^{[n+1]}=x^{[n]}x^{[n]},..$$.. If $$A$$ has a nontrivial algebra homomorphism $$\omega:A\to K$$, and if $$x^{[k+2]}=\omega (x)^{2^ k}x^{[k+1]}$$ then $$A$$ is called a $$k$$-th order Bernstein algebra.
The paper gives basic properties and examples of these algebras, and then goes on to consider a $$2m$$-ploid population with $$n+1$$ alleles. This gives rise to a nonassociative algebra $$G(n+1,2m,M)$$, and the following theorem establishes conditions, in terms of the matrix $$M$$, under which this algebra is a $$k$$-th order Bernstein algebra.
Theorem: (i) The algebra $$G(n+1,2,M)$$ is a kth-order Bernstein algebra if and only if $$M^ k=M^{k+1}$$, $$M^ j\neq M^{j+1}$$ $$(j=1,...,k-1)$$. (ii) The algebra $$G(n+1,2m,M)$$ $$(m>1)$$ is a $$k$$-th order Bernstein algebra for some $$k\leq n$$ if and only if 0 is a characteristic root of $$M$$ with multiplicity $$n$$.
Throughout, the genetic interpretation of the results is discussed.

##### MSC:
 17D92 Genetic algebras 92D10 Genetics and epigenetics