两性种群的演化代数
Evolution algebra of a bisexual population
摘要：
我们引入一个（进化）代数确定的系数的遗传的两性人口的代数结构常数。研究了该代数的基本性质。我们证明，这个代数是交换的（因此灵活），不结合，不一定权力结合。我们证明了两性种群的进化代数不是一个分压代数，而是一个双分压代数，因此它的平方是分压的。证明了该代数是一个巴拿赫代数。描述了演化代数的所有导子的集合。我们找到了种群的一个态成为对应于演化代数的演化算子的不动点或零点的必要条件。我们还建立了演化算子轨迹的极限点集的上估计。利用这些必要条件，我们对进化代数的一个特例（两性种群对“1”型的雌性和雄性都有偏好）进行了详细的分析。对于这样一种特殊情况，我们描述了幂等元的全集合和绝对幂零元的全集合。
英文摘要：
We introduce an (evolution) algebra identifying the coefficients of inheritance of a bisexual population as the structure constants of the algebra. The basic properties of the algebra are studied. We prove that this algebra is commutative (and hence flexible), not associative and not necessarily power associative. We show that the evolution algebra of the bisexual population is not a baric algebra, but a dibaric algebra and hence its square is baric. Moreover, we show that the algebra is a Banach algebra. The set of all derivations of the evolution algebra is described. We find necessary conditions for a state of the population to be a fixed point or a zero point of the evolution operator which corresponds to the evolution algebra. We also establish upper estimate of the limit points set for trajectories of the evolution operator. Using the necessary conditions we give a detailed analysis of a special case of the evolution algebra (bisexual population of which has a preference on type "1" of females and males). For such a special case we describe the full set of idempotent elements and the full set of absolute nilpotent elements.
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