Exact conditions for commutativity of projections with crossover

Standard Evolutionary processes can be understood as a stochastic process p = G p which propagates a parent population p at generation t to a parent population p via a stochastic operator G : Λ → Λ. Here, populations p ∈ Λ are distributions over a search space (or genotype space) G, and Λ is the space (or simplex) of such distributions. We address the details of the formalism and how G is given by crossover, mutation, and selection later.