Murray R. Bremner
University of Saskatchewan, Canada
Polynomial Identities for Bernstein Algebras of Simple Mendelian Inheritance.
Abstract: The Bernstein algebras D^n G_a of simple Mendelian inheritance are obtained by n-fold duplication of the gametic algebra G_a whose basis consists of the alleles A_1, ..., A_a with structure constants A_i A_j = (1/2) A_i + (1/2) A_j. We demonstrate the fundamental role played by the recombination identity R of degree 4 in the theory of Bernstein algebras. We simplify and generalize results of Bernad, Gonzalez, Martinez and Iltyakov by showing that every polynomial identity for G_a is a consequence of R and another identity Q of degree 4, and that every polynomial identity for the zygotic algebra D G_a is a consequence of R. We use computer algebra to determine the polynomial identities of degree <= 7 for the copular algebra D^2 G_a; in this case the values of R produce absolute zero-divisors.