标题： 一个进化代数链 显示英文标题

标题： A chain of evolution algebras

摘要： 我们引入了演化代数链的概念。 这种演化代数链的结构常数矩阵序列满足类似于 Chapman-Kolmogorov 方程的方程。 我们给出了一些例子 （时间齐次、时间非齐次、周期性等） 这样的链。 对于一个周期性的演化代数链，我们构造了一个连续的非同构演化代数集合，并证明相应的离散时间演化代数链在这个集合中是稠密的。 我们得到一个演化代数为 baric 的条件，并给出一个性质转变的概念。 对于几个演化代数链，我们描述了 baric 性质随时间的变化情况。 对于由两状态演化代数矩阵给出的演化代数链，我们定义了一个 baric 性质控制器函数，并在该控制器满足某些条件的情况下，证明该链几乎必然不是 baric（相对于 Lebesgue 测度）。 我们还构造了几乎必然为 baric 的演化代数链的例子。 我们证明存在一些演化代数链，如果在固定时间具有唯一（或无限多个）绝对幂零元，则在任何时间都具有唯一（或无限多个）绝对幂零元；也存在一些演化代数链不具有这种性质。 对于二维演化代数链的一个例子，我们给出了所有幂等元的完整集合，并证明对于某些参数值，幂等元的数量不随时间变化，但对于其他参数值存在临界时间$t_{c}$，当时间$t\geq t_{\rm c}$时该链只有两个幂等元，而当时间$t< t_{\rm c}$时该链有四个幂等元。 显示英文摘要

摘要： We introduce a notion of chain of evolution algebras. The sequence of matrices of the structural constants for this chain of evolution algebras satisfies an analogue of Chapman-Kolmogorov equation. We give several examples (time homogenous, time non-homogenous, periodic, etc.) of such chains. For a periodic chain of evolution algebras we construct a continuum set of non-isomorphic evolution algebras and show that the corresponding discrete time chain of evolution algebras is dense in the set. We obtain a criteria for an evolution algebra to be baric and give a concept of a property transition. For several chains of evolution algebras we describe the behavior of the baric property depending on the time. For a chain of evolution algebras given by the matrix of a two-state evolution we define a baric property controller function and under some conditions on this controller we prove that the chain is not baric almost surely (with respect to Lebesgue measure). We also construct examples of the almost surely baric chains of evolution algebras. We show that there are chains of evolution algebras such that if it has a unique (resp. infinitely many) absolute nilpotent element at a fixed time, then it has unique (resp. infinitely many) absolute nilpotent element any time; also there are chains of evolution algebras which have not such property. For an example of two dimensional chain of evolution algebras we give the full set of idempotent elements and show that for some values of parameters the number of idempotent elements does not depend on time, but for other values of parameters there is a critical time $t_{c}$ such that the chain has only two idempotent elements if time $t\geq t_{\rm c}$ and it has four idempotent elements if time $t< t_{\rm c}$