On the structure of the set of higher order spreading models

Abstract

We generalize some results concerning the classical notion of a spreading model for the spreading models of order ξ. Among them, we prove that the set SM w ξ (X) of the ξ-order spreading models of a Banach space X generated by subordinated weakly null F -sequences endowed with the prepartial order of domination is a semi-lattice. Moreover, if SM w ξ (X) contains an increasing sequence of length ω then it contains an increasing sequence of length ω 1 . Finally, if SM w ξ (X) is uncountable, then it contains an antichain of size the continuum.