Invertibility of linear $f$-order preservers

In this paper, we prove that monotonic linear transformations with respect to partial orders $\stackrel{*}{<}_f$, ${*}\kern1pt{<}_f$, ${<}\kern1pt{*}_f$, $\stackrel{\diamond}{<}_f$, $\stackrel{\sigma}{<}_f$ and $\stackrel{\sigma_1}{<}_f$ are invertible.