Phenotype algebra and the dominance map

In this article, we analyze the algebraic aspects of the phenotype of a specific gene with $n$ different alleles. We study the genetic realization algebra that models the probabilities of an offspring's phenotype given the phenotypes of its parents; we call this algebraic structure the phenotype algebra. We introduce a dominance map that assigns the probability distribution of an individual's possible phenotypes to a probability distribution of their possible genotypes. We use this dominance map, together with well-known concepts such as gametic algebra, the tensor product of spaces, and the adjoint of a linear map, to compute the phenotype algebra. We provide three simple examples to illustrate how the construction works. We also analyze the case of a gene where the dominance of one allele over another is incomplete. In this case, we obtain a Lotka-Volterra algebra associated with an antisymmetric matrix.