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 and 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 $\omega$-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.