Abstract: To further simplify the complex problems in information theory, the Shirshov algorithm was utilizedused to reduce the specific generation relations, providing an algebraic method a simplified algebraic method was given to prove that Markov chain’ sthe reverse chain and subchain of a Markov chain remainedwere still Markov chain. Based on characterizing Markov chains using semiringsOn the basis of describing Markov chain by using semiring, the characterization of Markov random field was furtherly studied. The Shirshov algorithm was employed to calculate the Gröbner-Shirshov basis of the generation relationship of Markov random field, obtaining the semiring Markov normal form generated by these relations.The Shirshov algorithm was used to calculate the Gröbner-Shirshov basis of Markov random field generation relationship, and the standard form of semiring generated by the generation relationship was obtained. From this standard form, an algebraic method was derived to determine whether random variables form a Markov random field. Additionally, standard forms for representing joint entropy, conditional entropy, and mutual information were obtained, as well as the standard representations of information measures such as joint entropy, conditional entropy, and mutual information. Finally, a specific Markov random field was used as an example to compute the Gröbner-Shirshov basis and normal form of its generating relations. It was concluded that the random variables X_1,X_2,X_3,X_4 , form the Markov random field if and only if for any p\in K_4,y_p=\theta , K_4=\left\\mathrm9,10,11\right\ .