F-evolution algebra

Abstract

We consider the evolution algebra of a free population generated by an F-quadratic stochastic operator. We prove that this algebra is commutative, not associative and necessarily power-associative. We show that this algebra is not conservative, not stationary, not genetic and not train algebra, but it is a Banach algebra. The set of all derivations of the F-evolution algebra is described. We give necessary conditions for a state of the population to be a fixed point or a zero point of the F-quadratic stochastic operator which corresponds to the F-evolution algebra. We also establish upper estimate of the !-limit set of the trajectory of the operator. For an F-evolution algebra of Volterra type we describe the full set of idempotent elements and the full set of absolute nilpotent elements.