Representation of Markov's chains functions over finite field based on stochastic matrix lumpability

A polynomial model of functions of lumped Markov's chains over finite field is represented. The authors represent a particular case of Markov's chains functions - the hidden Markov's models (HMM). A definition of the HMM stochastic equivalency was implemented based on stochastic matrix lumpability and asymptotic characteristics of HMM. The paper provides a polynomial representation of an equivalent HMM over the finite field GF(2
ⁿ
). It estimates a minimal order of the field GF(2
ⁿ
) for polynomial HMM representation, depending on the stochastic matrices implying vector size and possessing lumpability characteristic.