Lexicographic preferences representable by real-branching trees with countable height: a dichotomy result

The linear ordering R^\omega_\lex is the lexicographic linearization of the tree of R-valued functions defined on a finite initial segment of \omega and ordered by extension. We identify suitable notions of smallness and largeness for linear orderings that embed into R^\omega_\lex by using tree representations of chains. Specifically, small linear orderings are representable by inversely well-founded trees, and large linear orderings are representable by fully uncountably branching trees. We prove the rather surprising result that all linear orderings embeddable into R^\omega_\lex are either small or large. This fact sheds some light on the complicated structure of the linear ordering R^\omega_\lex, and can be useful in applications to utility theory and preference modeling.