On Lattice Properties of the Lorentz Spaces $L_{p,q}$

It is shown that the space $l_r$ is crudely finitely representable in the Lorentz space $L_{p,q}[0,1]$, $1<p\le q<\infty$, if and only if $r=p$ or $r=q$. To the best of the author's knowledge, this is the first example of a “natural” rearrangement-invariant space $E$ on $[0,1]$ such that the set of all numbers $r$ for which $l_r$ is crudely finitely representable in $E$ is not an interval of the real line.