  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.
