In order to investigate the relationship between Shannon information measure of random variables, scholars such as Yeung utilized information diagrams to explore the structured representation of information measures, establishing correspondences with sets. However, this method has limitations when studying information measures of five or more random variables. In this paper, we consider employing algebraic methods to study the relationship of information measures of random variables. By introducing a semiring generated by random variables, we establish correspondences between sets and elements of the semiring. Utilizing the Grobner-Shirshov basis, we present the structure of the semiring and its standard form. Furthermore, we delve into the structure of the semiring generated under Markov chain conditions (referred to as Markov semiring), obtaining its standard form.

翻译：为探究随机变量间香农信息测度的关系，Yeung等学者利用信息图研究信息测度的结构化表示，建立了与集合的对应关系。但该方法在研究五个及以上随机变量的信息测度时存在局限性。本文考虑采用代数方法研究随机变量信息测度的关系，通过引入由随机变量生成的半环建立集合与半环元素间的对应关系。借助Grobner-Shirshov基，我们给出了该半环的结构及其标准型。进一步地，深入研究了马尔可夫链条件下生成半环（称为马尔可夫半环）的结构，获得了其标准型。