Geodesic
Additive matrix maps that are monotone with respect to the orders induced by group inverse
Fundamentalʹnaâ i prikladnaâ matematika, Tome 17 (2012) no. 6, pp. 23-40

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We characterize additive maps on the matrix algebra over an arbitrary field with three or more elements that are monotone with respect to the $\overset\sharp\leq$- and $\overset{\mathrm{cn}}\leq$-orders and build some examples of nonadditive monotone maps. 
@article{FPM_2012_17_6_a1,
     author = {M. A. Efimov},
     title = {Additive matrix maps that are monotone with respect to the orders induced by group inverse},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {23--40},
     publisher = {mathdoc},
     volume = {17},
     number = {6},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2012_17_6_a1/}
}
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M. A. Efimov. Additive matrix maps that are monotone with respect to the orders induced by group inverse. Fundamentalʹnaâ i prikladnaâ matematika, Tome 17 (2012) no. 6, pp. 23-40. http://geodesic.mathdoc.fr/item/FPM_2012_17_6_a1/