Upward and Downward Runs on Partially Ordered Sets
نویسنده
چکیده
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,
منابع مشابه
Tripled partially ordered sets
In this paper, we introduce tripled partially ordered sets and monotone functions on tripled partiallyordered sets. Some basic properties on these new dened sets are studied and some examples forclarifying are given.
متن کاملBEST APPROXIMATION IN QUASI TENSOR PRODUCT SPACE AND DIRECT SUM OF LATTICE NORMED SPACES
We study the theory of best approximation in tensor product and the direct sum of some lattice normed spacesX_{i}. We introduce quasi tensor product space anddiscuss about the relation between tensor product space and thisnew space which we denote it by X boxtimesY. We investigate best approximation in direct sum of lattice normed spaces by elements which are not necessarily downwardor upward a...
متن کاملInterval fractional integrodifferential equations without singular kernel by fixed point in partially ordered sets
This work is devoted to the study of global solution for initial value problem of interval fractional integrodifferential equations involving Caputo-Fabrizio fractional derivative without singular kernel admitting only the existence of a lower solution or an upper solution. Our method is based on fixed point in partially ordered sets. In this study, we guaranty the existence of special kind of ...
متن کاملFixed point theorems for $alpha$-$psi$-contractive mappings in partially ordered sets and application to ordinary differential equations
In this paper, we introduce $alpha$-$psi$-contractive mapping in partially ordered sets and construct fixed point theorems to solve a first-order ordinary differential equation by existence of its lower solution.
متن کامل