Strong survival and extinction for multitype branching processes via a new order for generating functions

We consider general discrete-time multitype branching processes on a countable set $X$. According to these processes, a particle of type $x\in X$ generates a random number of children and chooses their type in $X$, not necessarily independently nor with the same law for different parent types. We introduce a new type of stochastic ordering of multitype branching processes, generalizing the germ order introduced by Hutchcroft in arXiv:2011.06402, which relies on the generating function of the process. We prove that given two multitype branching processes with law $\mathbf{\mu}$ and $\mathbf{\nu}$ respectively, with $\mathbf{\mu}\ge\mathbf{\nu}$, then in every set where there is survival according to $\\mathbf{\nu}$, there is survival also according to $\mathbf{\mu}$. Moreover, in every set where there is strong survival according to $\mathbf{\nu}$, there is strong survival also according to $\mathbf{\mu}$, provided that the supremum of the global extinction probabilities, for the $\mathbf{\nu}$-process, taken over all starting points $x$, is strictly smaller than 1. New conditions for survival and strong survival for inhomogeneous multitype branching processes are provided. We also extend a result of Moyal which claims that, under some conditions, the global extinction probability for a multitype branching process is the only fixed point of its generating function, whose supremum over all starting coordinates may be smaller than 1.