Disjointly strictly singular inclusions between variable Lebesgue spaces

Disjointly strictly singular inclusions between variable Lebesgue spaces $L^{p(\cdot )}(\mu )$ on finite measure are characterized. Suitable criteria in terms of the (bounded or unbounded) exponents are given. It is proved the equivalence of $L$-weak compactness (also called almost compactness) and disjoint strict singularity for variable Lebesgue space inclusions. For infinite measure any inclusion $L^{p(\cdot )}(\mu )\hookrightarrow L^{q(\cdot )}(\mu )$ is not disjointly strictly singular. No restrictions on the exponent are imposed.