zbMATH — the first resource for mathematics

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.

17D92 Genetic algebras
92D10 Genetics and epigenetics