Abstract: To further simplify complex problems in information theory, the Shirshov algorithm was employed to reduce the specific generation relations, through which simplified algebraic proofs were provided that the reversed chain and subchain of a Markov chain preserve the Markov property. Building upon the semiring-based characterization of Markov chains, the algebraic representation of Markov random fields was further explored. The Gröbner–Shirshov basis for the generating relations of Markov random fields was computed using the Shirshov algorithm, thereby obtaining the corresponding semiring Markov normal form. Based on this normal form, an algebraic criterion was established for determining whether random variables form a Markov random field, and standard representations were derived for information measures including joint entropy, conditional entropy, and mutual information. Finally, through a concrete example, the Gröbner–Shirshov basis and normal form of the generating relations for a Markov random field were computed, and it was proved that the random variables (X₁, X₂, X₃, X₄) constitute the given Markov random field if and only if for any p\in K_4,y_p=\theta , K_4=\left\\mathrm9,10,11\right\ .