Upward and Downward Runs on Partially Ordered Sets

We consider Markov chains on partially ordered sets that generalize the success-runs and remaining life chains in reliability theory. We find conditions for recurrence and transience and give simple expressions for the invariant distributions. We study a number of special cases, including rooted trees, uniform posets, and posets associated with positive semigroups. 1 Partially Ordered Sets 1.1 Preliminaries Suppose that (S, ) is a discrete partially ordered set. Recall that C ⊆ S is a chain if C is totally ordered under . We make the following assumptions: 1. There is a minimum element e. 2. For every x ∈ S, every chain in S from e to x is finite. Recall that y covers x if y is a minimal element of {t ∈ S : t ≻ x}. The covering graph (or Hasse graph) of (S, ) is the directed graph with vertex set S and edge set E = {(x, y) ∈ S : y covers x}. From the assumptions, it follows that for each x ∈ S, there is a (directed) path from e to x in the graph, and every such path is finite. For x ∈ S, let Ax = {y ∈ S : y covers x}, Bx = {w ∈ S : x covers w} That is, Ax is the set of elements immediately after x in the partial order, while Bx is the set of elements immediately before x in the partial order. Note that Ax could be empty or infinite. On the other hand, Be = ∅, but for x 6= e, Bx 6= ∅ since there is a path from e to x. An upward run chain on (S, ) is a Markov chain that, at each time, moves to a state immediately above the current state or all the way back down to e,