On the structure of variable exponent spaces

The first part of this paper surveys several results on the lattice structure of variable exponent Lebesgue function spaces (or Nakano spaces) $\text{lpv}$. In the second part strictly singular and disjointly strictly singular operators between spaces $\text{lpv}$ are studied. New results on the disjoint strict singularity of the inclusions $L^{p(\cdot )}(\mathrm{\Omega })\hookrightarrow L^{q(\cdot )}(\mathrm{\Omega })$ are given.