On the structure of the set of higher order spreading models

We generalize some results concerning the classical notion of a spreading model for the spreading models of order $\xi$. Among them, we prove that the set $SM_\xi^w(X)$ of the $\xi$-order spreading models of a Banach space $X$ generated by subordinated weakly null $\mathcal{F}$-sequences endowed with the pre-partial order of domination is a semi-lattice. Moreover, if $SM_\xi^w(X)$ contains an increasing sequence of length $\omega$ then it contains an increasing sequence of length $\omega_1$. Finally, if $SM_\xi^w(X)$ is uncountable, then it contains an antichain of size the continuum.