进化代数链
A chain of evolution algebras
摘要：
引入了演化代数链的概念。这一演化代数链的结构常数的矩阵序列满足Chapman-Kolmogorov方程的一个模拟。我们给出了几个例子（时间均匀，时间非均匀，周期性等）。这样的链条。对于演化代数的周期链，我们构造了一个非同构演化代数的连续集，并证明了相应的演化代数的离散时间链在该集中是稠密的。给出了演化代数是对称的一个判别准则，并给出了性质转移的概念。对于几个演化代数链，我们描述了baric属性依赖于时间的行为。对于由两态演化矩阵给出的演化代数链，我们定义了一个具有baric性质的控制器函数，并在一定条件下证明了该控制器几乎必然不具有baric性质（关于勒贝格测度）.我们还构造了演化代数的几乎必然对称链的例子。我们证明了存在演化代数链，使得如果它有一个唯一的（分别是）。无穷多个）绝对幂零元，则它有唯一的（分别是）无穷多个）绝对幂零元，也有演化代数链不具有这种性质。以二维演化代数链为例，给出了幂等元的全集合，并证明了对于某些参数值，幂等元的个数不依赖于时间，但对于另一些参数值，存在一个临界时间$t_{c}$，使得当时间为$t\geq t_{\rm c}$时，链只有两个幂等元，当时间为$t< t_{\rm c}$时，链有四个幂等元
英文摘要：
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 $t< t_{\rm c}$
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