$\newcommand{\om}{\omega} \newcommand{\Om}{\Omega}$ If $E$ is a distributive lattice with measurable binary operations $E\times E\ni(x,y)\mapsto x\wedge y\in E$ and $E\times E\ni(x,y)\mapsto x\vee y\in E$, then the "order statistics'' $X_{n:j}$ can be defined by the formula \begin{equation}\label{eq:wedge-vee} X_{n:j}(\om):=\bigwedge\Big\{\bigvee_{i\in J}X_i(\om)\colon J\in\binom{[n]}j\Big\} =\bigvee\Big\{\bigwedge_{i\in J}X_i(\om)\colon J\in\binom{[n]}{n+1-j}\Big\} \end{equation} for $j\in[n]:=\{1,\dots,n\}$ and $\om\in\Om$, with $\binom{[n]}j$ denoting the set of all subsets $J$ of the set $[n]$ such that the cardinality of $J$ is $j$; cf. formulas (1.2)--(1.4).
As explained in that paper in the paragraph right after (1.4), if $E$ is not a distributive lattice, then the two dual to each other natural expressions for $X_{n:j}$ in the above display may differ from each other, and thus no reasonable definition of $X_{n:j}$ will seem possible.