Suppose $X = (X_k)_{k=0}^\infty$ is a homogeneous Markov chain/process (for example on the state space $E = \lbrace 1, \dots, m\rbrace$). We can interpret the elements of the state space as "values". In the context of optimal stopping for example, we could set our gain/utility function to be the identity $g(x)=x$. Consider the problem of finding a stopping time $\tau$ that attains the supremum in $\sup_{0\le \tau \le N} \mathbb{E}[g(X_\tau)]$. Then in a sense, $\leq$ is a natural ordering of the states (with $m$ being the "best" state when $E = \lbrace 1, \dots, m\rbrace$).
I was wondering if there is a name for Markov Chains that preserve this ordering in the following sense. If $x \le y$ then $\mathbb{E}[X_{n+1} | X_n = x] \le \mathbb{E}[X_{n+1} | X_n = y]$.
Many stochastic processes satisfy this property, for example the random walk (with absorbing boundaries). If we consider an analog for Markov processes then any Lévy process satisfies the property because of the properties of independent increments and stationary increments. In addition any Martingale satisfies the property. But those conditions are not necessary, as for example the Ornstein-Uhlenbeck process also satisfies my property.
The property seems relevant in the context of optimal stopping. Is there a name for this property, or similar properties?