关于代数集的马尔可夫问题的一致解

群中的无条件τ-闭集与τ-代数集

Unconditionally tau-Closed and tau-Algebraic Sets in Groups

定义了群中的无条件$τ$ -闭集和$τ$ -代数集族，它们是Markov定义的无条件闭集和无条件代数集的自然推广.找到了这两个族重合的一个充分条件，特别证明了这两个族在任意基数群中至多重合$τ$。这一结果推广了马尔可夫定理的重合无条件封闭和代数集在一个可数群（众所周知，他们可能是不同的不可数群）和Podewski定理的拓扑性的任何ungebunden群。

一类所有无条件闭集都是代数的群

A Class of Groups in Which All Unconditionally Closed Sets are Algebraic

证明了在可数群的直积的某些子群中，作为无条件闭集的性质与作为代数集的性质是一致的。特别地，这些性质在所有的阿贝尔群中一致。

Uniformization stable Markov models and their Jordan algebraic structure

We provide a characterisation of the continuous-time Markov models where the Markov matrices from the model can be parameterised directly in terms of the associated rate matrices (generators). That is, each Markov matrix can be expressed as the sum of the identity matrix and a rate matrix from the model. We show that the existence of an underlying Jordan algebra provides a sufficient condition, which becomes necessary for (so-called) linear models. We connect this property to the well-known uniformization procedure for continuous-time Markov chains by demonstrating that the property is equival…

The Markov-Zariski topology of an abelian group

According to Markov, a subset of an abelian group G of the form {x in G: nx=a}, for some integer n and some element a of G, is an elementary algebraic set; finite unions of elementary algebraic sets are called algebraic sets. We prove that a subset of an abelian group G is algebraic if and only if it is closed in every precompact (=totally bounded) Hausdorff group topology on G. The family of all algebraic subsets of an abelian group G forms the family of closed subsets of a unique Noetherian T_1 topology on G called the Zariski, or verbal, topology of G. We investigate the properties of this …

Algebraic sets with fully characteristic radicals

We obtain a necessary and sufficient condition for an algebraic set in a group to have a fully characteristic radical. As a result, we see that if the radical of a system of equation $S$ over a group $G$ is fully characteristic, then there exists a class $\mathfrak{X}$ of subgroups of $G$ such that elements of $S$ are identities of $\mathfrak{X}$.

虚拟阿贝尔群代数集的多变量增长级数的有理性

Rationality of the multivariate growth series for algebraic sets of virtually abelian groups

本文证明了虚拟阿贝尔群代数集的多变量相对增长级数具有有理性，证实了 Evetts 和 Levine 的猜想。