The combinational structure of non-homogeneous Markov chains with countable states

Let P(s,t) denote a non-homogeneous continuous parameter Markov chain with countable state space E and parameter space [a,b], −∞ 0}. It is shown in this paper that R(s,t) is reflexive, transitive, and independent of (s,t), shomogeneity condition holds. It is also shown that the relation R(s,t), unlike in the finite state space case, cannot be expressed even as an infinite (countable) product of reflexive transitive relations for certain non-homogeneous chains in the case when E is infinite.