= 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.
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