Necessary embedding conditions for state-wise monotone Markov chains.

In previous work, the embedding problem is examined within the entire set of discrete-time Markov chains. However, for several phenomena, the states of a Markov model are ordered categories and the transition matrix is state-wise monotone. The present paper investigates the embedding problem for the specific subset of state-wise monotone Markov chains. We prove necessary conditions on the transition matrix of a discrete-time Markov chain with ordered states to be embeddable in a state-wise monotone Markov chain regarding time-intervals with length 0.5: A transition matrix with a square root within the set of state-wise monotone matrices has a trace at least equal to 1.