Non-Associativity of two-dimensional b-Bistochastic-Volterra genetic algebra

The development in genetic algebra, including baric, evolution, Bernstein, train, stochastic, and others, has made significant contributions to the theory of population genetics. Bernstein’s research on exploring evolution operators was the original works on population genetics. This concept is being generalized by the theory quadratic stochastic operators (QSOs) in which induces an algebraic structure on the vector space ℝⁿ called genetic algebras. Unlike linear operator, exploring arbitrary QSOs for any finite dimension presents a challenging problem, therefore a viable strategy to tackle this issue is to introduce classes of QSOs. Hence, this study focuses specifically on a class of QSOs namely b−Bistochastic Volterra QSOs, simply written as bV-QSOs. This operator establishes a genetic algebra on ℝⁿ × ℝⁿ which become the primary focus of our study. To the best of the authors’ knowledge, this is the first time in the literature to define such kind of genetic algebras. Further, we investigate the associativity property of this algebra, demonstrating that two-dimensional bV genetic algebra fails to be associative.