  This paper is motivated by examples from stochastic reaction network theory.
The $Q$-matrix of a stochastic reaction network can be derived from the
reaction graph, an edge-labelled directed graph encoding the jump vectors of an
associated continuous time Markov chain on the invariant space
$\mathbb{N}^d_0$. An open question is how to decompose the space
$\mathbb{N}^d_0$ into neutral, trapping, and escaping states, and open and
closed communicating classes, and whether this can be done from the reaction
graph alone. Such general continuous time Markov chains can be understood as
natural generalizations of birth-death processes, incorporating multiple
different birth and death mechanisms. We characterize the structure of
$\mathbb{N}^d_0$ imposed by a general $Q$-matrix generating continuous time
Markov chains with values in $\mathbb{N}^d_0$, in terms of the set of jump
vectors and their corresponding transition rate functions. Thus the setting is
not limited to stochastic reaction networks. Furthermore, we define structural
equivalence of two $Q$-matrices, and provide sufficient conditions for
structural equivalence. Examples are abundant in applications. We apply the
results to stochastic reaction networks, a Lotka-Volterra model in ecology, the
EnvZ-OmpR system in systems biology, and a class of extended branching
processes, none of which are birth-death processes.
