 A function of a Markov chain need not be Markov

A function of the current state of a Markov chain can discard information needed to predict its next value. A sufficient condition preserving the Markov property is strong lumpability: all original states with the same recorded value must give the same transition probabilities to every recorded class.

For the two-column model in growth rate of two-column sticky deposition, the pair of column heights is Markov, but its maximum alone is not. Starting empty, the maximum-height histories $(1,1,2)$ and $(1,2,2)$ both end at height two. The former leaves unequal columns, so the next height increases with probability $1/2$; the latter leaves equal columns, so it increases with probability one. Each history has positive probability. Their different predictions at the same time and current value violate the Markov property. By contrast, the absolute height difference retains exactly the information needed for its own transitions.